# New Classes of Binary Sequences with High Merit Factor

Miroslav Dimitrov This work has been partially supported by the Bulgarian
National Science Fund under contract number DH 12/8, 15.12.2017.M. Dimitrov is
with the Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, Sofia, Bulgaria (email:mirdim@math.bas.bg).

###### Abstract

The Merit Factor (MF) measure was first introduced by Golay in 1972. Sequences
possessing large values of MF are of great interest to a rich list of
disciplines - from physics and chemistry to digital communications, signal
processing, and cryptography. Throughout the last half-century, manifold
approaches and strategies were proposed for finding such sequences. Referenced
as one of the most difficult optimization problems, Golay wrote that it is a
”challenging and charming problem”. His publications on this problem spanned
more than 20 years. Golay himself introduced one beneficial class of
sequences, called skew-symmetric sequences, or finite binary sequences with
odd lengths with alternate autocorrelation values equal to 0. Their sieving
construction greatly reduces the computational efforts of finding binary
sequences with odd lengths and high MF. Having this in mind, the majority of
papers to be found in the literature are focused solely on this class,
preferring them over binary sequences with even lengths. In this work, a new
class of finite binary sequences with even lengths with alternate
autocorrelation absolute values equal to 1 is presented. We show that the MF
values of the new class are closely related to the MF values of adjacent
classes of skew-symmetric sequences. We further introduce new sub-classes of
sequences using the partition number problem and the notion of potentials,
measured by helper ternary sequences. Throughout our experiments, MF records
for binary sequences with many lengths less than 225, and all lengths greater
than 225, are discovered. Binary sequences of all lengths, odd or even, less
than $2^{8}$ and with MF $>8$, and all lengths, odd or even, less than $2^{9}$
and with MF $>7$, are now revealed.

###### Index Terms:

Binary Sequences, Algorithms, Merit Factor Problem

##  I Introduction

Finding binary sequences whose aperiodic autocorrelation characteristics are
collectively small according to some pre-defined criteria is a famous and
well-studied problem. Two such measures are the Peak Sidelobe Level (PSL) and
the Merit Factor (MF) value, which was first introduced by Golay in 1972 [1].
Additionally, prior to Golay’s definition, Littlewood [2] also studied the
norms of polynomials with $\pm$1 coefficients on the unit circle of the
complex plane, which appears to be the same problem.

Golay’s publications reveal a dedication to the merit factor problem for
nearly twenty years (surveyed in [3]). Since then, a significant number of
possible constructions of binary sequences with high merit factors were
published. Near-optimal and optimal candidates are found by using heuristic
search methods for longer lengths or a more direct approach, as the exhaustive
search method, for smaller problem spaces.

The optimal merit factors for all binary sequences with lengths $n\leq 60$ are
presented in [4]. Twenty years later, the list of optimal merit factors was
extended to $n\leq 66$ [5]. The two largest known merit factor values are 14.1
and 12.1 for lengths respectively 13 and 11. Those binary sequences are
comprised of the Barker sequences [6]. The merit factor problem was also
attacked by various algorithms, such as the branch and bound algorithm [5],
stochastic search algorithms like tabu search [7], memetic algorithm combined
with tabu search [8], as well as evolutionary and genetic algorithms [9][10].
However, since the search space grows like $2^{n}$, the difficulty of finding
long binary sequences with near-optimal MF significantly increases.

A reasonable strategy for finding longer binary sequences with near-optimal
merit factor is to introduce some restriction on the sequences’ structure,
which could decrease the size the of the search space. A well-studied
restriction on the structure of the sequence has been defined by the skew-
symmetric binary sequences, which were introduced by Golay [1]. Having a
binary sequence $(b_{0},b_{1},\cdots,b_{2l})$ of odd length $n=2l+1$, the
restriction is defined by $b_{l+i}=(-1)^{i}b_{l-i}\text{ for }i=1,2,\cdots,l.$
Golay observed that odd length Barker [6] sequences are skew-symmetric and the
idea of binary sequences’ sieving was proposed. This proposition was
accompanied by the observation that the alternate aperiodic autocorrelation
values of a skew-symmetric sequence are equal to 0. The optimal merit factors
for all skew-symmetric sequences of length $n\leq 59$ were given by Golay
himself [11]. Later, the optimal merit factors for skew-symmetric sequences
with lengths $n\leq 69$ [12] and $n\leq 71$ [9] were also revealed, while the
optimal skew-symmetric solutions for $n\leq 89$ and $n\leq 119$ were
discovered in respectively [13] and [5].

In the case of skew-symmetric binary sequences, Golay suggested [14] that only
one or two elements should be changed at a given optimization step. In case a
new candidate has a better merit factor, the search method accepts it as a new
current state and continues the optimization process. Having this in mind, a
strategy of how to choose a new sequence when no acceptable neighbor sequence
exists is another nuance of the search routine.

Besides the exhaustive search approach, near-optimal results regarding skew-
symmetric binary sequences’ high merit factor problem are achieved by
[8][15][16][17]. In [8], the authors introduced a memetic algorithm with an
efficient method to recompute the characteristics of a given binary sequence
$L^{\prime}$, such that $L^{\prime}$ is one flip away from $L$, and assuming
that some products of elements from $L$ have been already stored in memory -
the tau table $\scalerel*{\tau}{T}(S)$ was introduced. Later, in [15] the
principle of self-avoiding walk [18] was considered. Then, in [16] an
algorithm called xLastovka was presented and the concept of a priority queue
was introduced. However, despite the rich results regarding the skew-symmetric
binary sequences, the search for binary sequences with even lengths and high
MF was scarcely researched. This is not surprising, since the sieving proposed
by Golay is applicable to odd-length sequences only.

More details regarding the history of the problem, as well as the current
achievements in the theoretical aspect of it, could be found in surveys [19],
[20] and [3]. Considering more theoretical results, in [21] the authors
construct an infinite family of binary sequences whose asymptotic merit factor
is conjectured to be greater than 6.34. Nevertheless, as the authors stated in
the abstract of their work, “… the numerical experimentation that led to this
construction is a significant part of the story.“

In this work, motivated by the absence of computationally efficient sieving
for binary sequences with even length and high merit factor values, several
new classes of binary sequences are proposed. In Chapter III, we start with
the definition of a class of finite binary sequences, called pseudo-skew-
symmetric, with alternate auto-correlation absolute values equal to one. The
class is defined by using sieving suitable for even-length binary sequences.
Then, by using some mathematical observations, we show how state-of-the-art
algorithms for searching skew-symmetric binary sequences with high merit
factor and length $2n+1$ could be easily converted to algorithms searching
pseudo-skew-symmetric binary sequences with high merit factor and lengths $2n$
or $2n+2$. More importantly, this conversion does not degrade the performance
of the modified algorithm.

In Chapter IV, by using number partitions [22], additional sieving strategy
for both skew-symmetric and pseudo-skew-symmetric sequences is proposed. A
method of finding sub-classes of binary sequences with high MF is further
discussed. The final algorithm is presented in Chapter V. The experiments in
Chapter VI revealed that the classes defined in this work are highly
promising. By using a single mid-range computer, we were able to improve all
records for skew-symmetric binary sequences with lengths above 225, which were
recently reached by various algorithms and a supercomputer grid. We further
revealed that binary sequences with even or odd length $n$, for $n\leq 2^{8}$,
and with merit factor strictly greater than 8, do exist, while binary
sequences with even or odd length $n$, for $n\leq 2^{9}$ and with a merit
factor strictly greater than 7 are also revealed.

Finally, we demonstrate the efficiency of the proposed algorithm by launching
it on two extremely hard search spaces of binary sequences of lengths 573 and
1009. The choice of those two specific lengths is motivated by the
approximation numbers given in [15], Figure 7, presented during a discussion
of how much time the state-of-the-art stochastic solver lssOrel_8 will need to
reach binary sequences with the aforementioned lengths and merit factors close
to $6.34$. It was estimated that finding solutions with merit factor of 6.34
for a binary sequence with length 573 requires around 32 years, while for
binary sequences with length 1009, the average runtime prediction to reach the
merit factor of 6.34 is $46774481153$ years. By using the proposed in this
work algorithm, we were able to reach such candidates within several hours.

##  II Preliminaries

We denote as $B=(b_{0},b_{1},\cdots,b_{n-1})$ the binary sequence with length
$n>1$, such that $b_{i}\in\\{-1,1\\},0\leq i\leq n-1$. Throughout this paper,
the binary sequence could be represented with the concatenation symbol as
well, i.e. $B=b_{0}||b_{1}||\cdots||b_{n-1}$.

The aperiodic autocorrelation function of $B$ is given by

$C_{u}(B)=\sum_{j=0}^{n-u-1}b_{j}b_{j+u},\ \ for\ u\in\\{0,1,\cdots,n-1\\}.$

We define $C_{u}(B)$ for $u\in\\{1,\cdots,n-1\\}$ as a sidelobe level.
$C_{0}(B)$ is called the mainlobe.

The enery $\mathbb{E}(B)$ of $B$ is defined as

$\mathbb{E}(B)=\sum_{u=1}^{n-1}C_{u}(B)^{2},$

while the merit factor, or MF, $\mathbb{MF}(B)$ of $B$ is defined as

$\mathbb{MF}(B)=\frac{n^{2}}{2\mathbb{E}(B)}.$

For convenience, we will denote $C_{n-i-1}(B)$ by $\hat{C}_{i}(B)$.

##  III Pseudo-Skew-Symmetric Sequences and Their Energy

We consider a skew-symmetric binary sequence defined by a binary sequence
$B=\left(b_{0},b_{1},\cdots,b_{n-1}\right)$ with an odd length $n=2l+1$.

In this paper, for convenience, we will denote with $S_{B}$ the array of the
sidelobes of $B$:

$S_{B}=\left[\hat{C}_{0}(B),\hat{C}_{1}(B),\cdots,\hat{C}_{n-2}(B),\hat{C}_{n-1}(B)\right],$

where $\hat{C}_{n-i-1}(B)=C_{i}(B)$, for $i\in\\{0,1,\cdots,n-1\\}$. Thus,

$\hat{C}_{i}(B)=C_{n-i-1}(B)=\sum_{j=0}^{n-(n-i-1)-1}b_{j}b_{j+(n-i-1)}=\sum_{j=0}^{i}b_{j}b_{j+n-i-1},\
\ for\ i\in\\{0,1,\cdots,n-1\\}.$

The $i$-th element of a given array $S$ is denoted as $S[i]$. For practical
reasons, we will define the first index of an array as 0, not 1. For example,

$S_{B}[0]=\hat{C}_{0}(B)=C_{n-1}(B).$

Since $B$ is a skew-symmetric binary sequence, the following properties hold:

  * •

$S_{B}[i]=0$, for odd values of $i$.

  * •

$B[l-i]={(-1)}^{i}B[l+i]$

Having this in mind, the array of sidelobes $S_{B}$ could be represented as
follows:

$S_{B}=\left[\hat{C}_{0}(B),0,\hat{C}_{2}(B),0,\cdots,0,\hat{C}_{n-3}(B),0,\hat{C}_{n-1}(B)\right].$

This property is beneficial for minimizing the energy $E(B)$ of the sequence.
Indeed,

$\mathbb{E}(B)=\sum_{u=0}^{n-2}\hat{C}_{u}(B)^{2}=\sum_{u=0,u_{even}}^{n-2}\hat{C}_{u}(B)^{2}+\sum_{u=0,u_{odd}}^{n-2}\hat{C}_{u}(B)^{2}=\sum_{u=0,u_{even}}^{n-2}\hat{C}_{u}(B)^{2}.$

Unfortunately, such an approach is not suitable for binary sequences with even
lengths. However, it is possible to construct an even-length binary sequence
with alternating absolute values equal to 1.

######  Definition 1 (Pseudo-Skew-Symmetric Binary Sequence)

We call a given sequence $P=a||X=Y||b$ a pseudo-skew-symmetric binary
sequence, if either $X$ or $Y$ are skew-symmetric binary sequences, for some
$a\in\\{-1,1\\}$ or $b\in\\{-1,1\\}$.

###### Proposition 1

The sidelobes array of pseudo-skew-symmetric binary sequences consists of
alternating $\pm$ ones.

###### Proof:

Let us denote the pseudo-skew-symmetric binary sequence as $P$. By definition,
$P$ could be represented as $a||A$ or $B||b$, for some skew-symmetric binary
sequences $A$ or $B$.

If $P=B||b$, for some skew-symmetric binary sequence
$B=\left(b_{0},b_{1},\cdots,b_{n-1}\right)$, then
$P=\left(b_{0},b_{1},\cdots,b_{n-1},b_{n}\right)$, where $b_{n}=b$.

Thus,

$\hat{C}_{i}(P)=\sum_{j=0}^{i}b_{j}b_{j+n-i},\ \ for\ i\in\\{0,1,\cdots,n\\}.$

Therefore, the sidelobes of $P$ could be further simplified:

$\hat{C}_{i}(P)=\sum_{j=0}^{i}b_{j}b_{j+n-i}=\sum_{j=0}^{i-1}b_{j}b_{j+n-i}+b_{i}b_{i+n-i}=\sum_{j=0}^{i-1}b_{j}b_{j+n-i}+b_{i}b_{n}=\hat{C}_{i-1}(B)+b_{i}b_{n}.$

The last substitution arises from the definition of the sidelobe array:

$\hat{C}_{i-1}(B)=\sum_{j=0}^{i-1}b_{j}b_{j+n-(i-1)-1}=\sum_{j=0}^{i-1}b_{j}b_{j+n-i}$

The sidelobe array of $P$, denoted as $S_{P}$, could be then simplified:

$\begin{split}S_{P}&=\left[\hat{C}_{0}(P),\hat{C}_{1}(P),\cdots,\hat{C}_{n-1}(P),\hat{C}_{n}(P)\right]=\\\
&=\left[\hat{C}_{0}(P),\hat{C}_{0}(B)+b_{1}b_{n},\hat{C}_{1}(B)+b_{2}b_{n},\cdots,\hat{C}_{n-2}(B)+b_{n-1}b_{n},\hat{C}_{n-1}(B)+b_{n}b_{n}\right]=\\\
&=\left[b_{0}b_{n},\hat{C}_{0}(B)+b_{1}b_{n},b_{2}b_{n},\cdots,b_{n-1}b_{n},\hat{C}_{n-1}(B)+b_{n}b_{n}\right]\\\
\end{split}$ (1)

Note that $\hat{C}_{x}(B)=0$, for odd values of $x$. Since
$b_{i}\in\\{-1,1\\}$, $\hat{C}_{x}(P)=b_{x}b_{n}=\pm 1$, for even values of
$x$, which completes the proof of the first case, or more formally:

$S_{P}=\left[\pm 1,\hat{C}_{0}(B)+b_{1}b_{n},\pm 1,\cdots,\pm
1,\hat{C}_{n-1}(B)+b_{n}b_{n}\right].$

Let us consider the second case. If $P=a||A$, then $P^{rev}=A^{rev}||a$ will
posses the same sidelobes array as $P$, where $A^{rev}$ denotes the reversed
version of a given binary sequence $A$. Since $A$ is skew-symmetric sequence,
$A^{rev}$ is skew-symmetric as well. Thus, by applying the first case, we
have:

$S_{P}\equal S_{P^{rev}}=\left[\pm 1,\hat{C}_{0}(A^{rev})+a_{1}a_{n},\pm
1,\cdots,\pm 1,\hat{C}_{n-1}(A^{rev})+a_{n}a_{n}\right],$

where $A^{rev}=\left(a_{0},a_{1},\cdots,a_{n-1},a_{n}\right)$ and $a_{n}=a$,
which completes the proof.

∎

This property is beneficial for the energy $E(P)$ of the pseudo-skew-symmetric
binary sequence $P$. Indeed,

$\mathbb{E}(P)=\sum_{u=0}^{n-1}\hat{C}_{u}(P)^{2}=\sum_{u=0,u_{even}}^{n-1}\hat{C}_{u}(P)^{2}+\sum_{u=0,u_{odd}}^{n-1}\hat{C}_{u}(P)^{2}=\sum_{u=0,u_{even}}^{n-1}{\pm
1}^{2}+\sum_{u=0,u_{odd}}^{n-1}\hat{C}_{u}(P)^{2}=\lfloor\frac{n}{2}\rfloor+\sum_{u=1,u_{odd}}^{n}\hat{C}_{u}(P)^{2}.$

The following property allows us to convert an existing algorithm for
searching skew-symmetric binary sequences with high merit factor to an
algorithm searching pseudo-skew-symmetric binary sequences and high merit
factor.

###### Proposition 2

Given a skew-symmetric binary sequence
$B=\left(b_{0},b_{1},\cdots,b_{n-1}\right)$ with sidelobes array

$S_{B}=\left[\hat{C}_{0}(B),\hat{C}_{1}(B),\cdots,\hat{C}_{n-2}(B),\hat{C}_{n-1}(B)\right],$

the following property holds:

$\mathbb{E}(P)=\mathbb{E}(B)+n+2b_{n}\delta,$

where $P$ is the pseudo-skew-symmetric sequence $B||b_{n}$ and
$\delta=\sum_{u=0,u_{even}}^{n-2}{\hat{C}_{u}(B)b_{u+1}}$.

###### Proof:

Using the result from the previous proposition proof we have:

$S_{P}=\left[\pm 1,\hat{C}_{0}(B)+b_{1}b_{n},\pm 1,\cdots,\pm
1,\hat{C}_{n-1}(B)+b_{n}b_{n}\right]$

By using the definition of energy of a binary sequence we have:

$\begin{split}\mathbb{E}(P)-\mathbb{E}(B)=&\sum_{u=0}^{n-1}\hat{C}_{u}(P)^{2}-\sum_{u=0}^{n-2}\hat{C}_{u}(B)^{2}=1+\sum_{u=1}^{n-1}\hat{C}_{u}(P)^{2}-\sum_{u=0}^{n-2}\hat{C}_{u}(B)^{2}=1+\sum_{u=0}^{n-2}{\hat{C}_{u+1}(P)^{2}-\hat{C}_{u}(B)^{2}}=\\\
&=1+\sum_{u=0,u_{even}}^{n-2}{\hat{C}_{u+1}(P)^{2}-\hat{C}_{u}(B)^{2}}+\sum_{u=0,u_{odd}}^{n-2}{\hat{C}_{u+1}(P)^{2}-\hat{C}_{u}(B)^{2}}=\\\
&=1+\sum_{u=0,u_{even}}^{n-2}{\left(\hat{C}_{u}(B)+b_{u+1}b_{n}\right)^{2}-\hat{C}_{u}(B)^{2}}+\sum_{u=0,u_{odd}}^{n-2}{\pm
1^{2}-0^{2}}=\\\
&=1+\sum_{u=0,u_{even}}^{n-2}{\left(\hat{C}_{u}(B)+b_{u+1}b_{n}\right)^{2}-\hat{C}_{u}(B)^{2}}+\lfloor\frac{n-1}{2}\rfloor=\\\
&=1+\sum_{u=0,u_{even}}^{n-2}{2\hat{C}_{u}(B)b_{u+1}b_{n}+\left(b_{u+1}b_{n}\right)^{2}}+\lfloor\frac{n-1}{2}\rfloor=\\\
&=1+\sum_{u=0,u_{even}}^{n-2}{2\hat{C}_{u}(B)b_{u+1}b_{n}}+\sum_{u=0,u_{even}}^{n-2}{\left(b_{u+1}b_{n}\right)^{2}}+\lfloor\frac{n-1}{2}\rfloor=\\\
&=1+\sum_{u=0,u_{even}}^{n-2}{2\hat{C}_{u}(B)b_{u+1}b_{n}}+\lfloor\frac{n-1}{2}\rfloor+\lfloor\frac{n-1}{2}\rfloor=n+2b_{n}\sum_{u=0,u_{even}}^{n-2}{\hat{C}_{u}(B)b_{u+1}}.\end{split}$
(2)

∎

The last property is of significant importance when converting an algorithm
searching for skew-symmetric binary sequences, denoted as $\mathscr{A}$, to an
algorithm searching for pseudo-skew-symmetric binary sequences $\mathscr{B}$
and a high merit factor. Indeed, despite the complexity of algorithm
$\mathscr{A}$ we can decompose it to a tape
$\cdots||\mathbb{L}_{1}||\cdots||\mathbb{L}_{2}||\cdots||\mathbb{L}_{n}||\cdots$,
where $\mathbb{L}_{i}$ are stages of $\mathscr{A}$, where better candidates
could be announced. They are known as local optimums in heuristic search
literature. We could easily replace $\mathbb{L}_{i}$ with
$\mathbb{L}_{i}||\mathbb{T}_{i}$, where $\mathbb{T}_{i}$ is a simple routine
with memory and time complexity of $O(n)$, which calculates the pseudo-skew-
symmetric sequences $L_{i}||1$ and $L_{i}||-1$ merit factors, where $L_{i}$ is
the current best candidate. It should be noted that
$\mathscr{B}=\cdots||\mathbb{L}_{1}||\mathbb{T}_{1}||\cdots||\mathbb{L}_{2}||\mathbb{T}_{2}||\cdots||\mathbb{L}_{n}||\mathbb{T}_{n}||\cdots$
does not interfere with the normal work of $\mathscr{A}$ by design.
Furthermore, since those linear time complexity checkups are initiated on
local optimums only, the delay of $\mathscr{B}$ compared to $\mathscr{A}$
caused by the additional instructions $\mathbb{T}_{i}$ is negligible.

We could further extend the search of highly-competitive pseudo-skew-symmetric
sequences by the following observation:

###### Proposition 3

Given a skew-symmetric binary sequence $B=b_{0}||B^{\prime}||b_{n-1}$ both
binary sequences $b_{0}||B^{\prime}$ and $B^{\prime}||b_{n-1}$ are pseudo-
skew-symmetric.

###### Proof:

From the main property of the skew-symmetric sequences follows that
$B^{\prime}$ is skew-symmetric as well. Thus, the pseudo-skew-symmetry of
$b_{0}||B^{\prime}$ and $B^{\prime}||b_{n-1}$ follows directly from definition
1. ∎

###### Proposition 4

Given a skew-symmetric binary sequence
$B=\left(b_{0},b_{1},\cdots,b_{n-1}\right)=b_{0}||B^{\prime}||b_{n-1}$ with
sidelobes array

$S_{B}=\left[\hat{C}_{0}(B),\hat{C}_{1}(B),\cdots,\hat{C}_{n-2}(B),\hat{C}_{n-1}(B)\right],$

the following property holds:

$\mathbb{E}(P)=\mathbb{E}(B)+n-3+2b_{n-1}\delta,$

where $P$ is the pseudo-skew-symmetric sequence $b_{0}||B^{\prime}$ and
$\delta=\sum_{u=1,u_{even}}^{n-2}-\hat{C}_{u}(B)b_{u}$.

###### Proof:

We have $B=\left(b_{0},b_{1},\cdots,b_{n-1}\right)$ and
$P=\left(b_{0},b_{1},\cdots,b_{n-2}\right)$. Furthermore,

$\hat{C}_{i}(P)=\sum_{j=0}^{i}b_{j}b_{j+n-2-i},\ \ for\
i\in\\{0,1,\cdots,n-2\\}.$

Decomposing $\hat{C}_{i}(B)$ reveals the following:

$\hat{C}_{i}(B)=\sum_{j=0}^{i}b_{j}b_{j+n-1-i}=\sum_{j=0}^{i-1}b_{j}b_{j+n-1-i}+b_{i}b_{i+n-1-i}=b_{i}b_{n-1}+\hat{C}_{i-1}(P)$

In other words, $\hat{C}_{i-1}(P)=\hat{C}_{i}(B)-b_{i}b_{n-1}$. Thus, by using
the sidelobes array of $B$,

$S_{B}=\left[\hat{C}_{0}(B),0,\hat{C}_{2}(B),0,\cdots,0,\hat{C}_{n-3}(B),0,\hat{C}_{n-1}(B)\right],$

we could represent the sidelobes array of $P$:

$\begin{split}S_{P}&=\left[\hat{C}_{0}(P),\hat{C}_{1}(P),\hat{C}_{2}(P),\cdots,\hat{C}_{n-2}(P)\right]=\\\
&=\left[\hat{C}_{1}(B)-b_{1}b_{n-1},\hat{C}_{2}(B)-b_{2}b_{n-1},\hat{C}_{3}(B)-b_{3}b_{n-1},\cdots,\hat{C}_{n-1}(B)-b_{n-1}b_{n-1}\right]=\\\
\end{split}$ (3)

By using the definition of energy of a binary sequence we have:

$\begin{split}\mathbb{E}(P)-\mathbb{E}(B)&=\sum_{u=0}^{n-3}\hat{C}_{u}(P)^{2}-\sum_{u=0}^{n-2}\hat{C}_{u}(B)^{2}=\sum_{u=0}^{n-3}\hat{C}_{u}(P)^{2}-\left(1+\sum_{u=1}^{n-2}\hat{C}_{u}(B)^{2}\right)=-1+\sum_{u=1}^{n-2}\hat{C}_{u-1}(P)^{2}-\hat{C}_{u}(B)^{2}=\\\
&=-1+\sum_{u=1,u_{even}}^{n-2}\hat{C}_{u-1}(P)^{2}-\hat{C}_{u}(B)^{2}+\sum_{u=1,u_{odd}}^{n-2}\hat{C}_{u-1}(P)^{2}-\hat{C}_{u}(B)^{2}=\\\
&=-1+\sum_{u=1,u_{even}}^{n-2}\left(\hat{C}_{u}(B)-b_{u}b_{n-1}\right)^{2}-\hat{C}_{u}(B)^{2}+\sum_{u=1,u_{odd}}^{n-2}{\pm
1^{2}-0^{2}}=\\\
&=-1+\sum_{u=1,u_{even}}^{n-2}\left(-2\hat{C}_{u}(B)b_{u}b_{n-1}+\left(b_{u}b_{n-1}\right)\right)^{2}+\lfloor\frac{n-2}{2}\rfloor=\\\
&=-1+\sum_{u=1,u_{even}}^{n-2}-2\hat{C}_{u}(B)b_{u}b_{n-1}+\sum_{u=1,u_{even}}^{n-2}\left(b_{u}b_{n-1}\right)^{2}+\lfloor\frac{n-2}{2}\rfloor=\\\
&=-1+\sum_{u=1,u_{even}}^{n-2}-2\hat{C}_{u}(B)b_{u}b_{n-1}+\lfloor\frac{n-2}{2}\rfloor+\lfloor\frac{n-2}{2}\rfloor=n-3+2b_{n-1}\sum_{u=1,u_{even}}^{n-2}-\hat{C}_{u}(B)b_{u}\end{split}$
(4)

∎

The last property further enhances the power of the algorithm. Thus now we can
modify each algorithm $\mathscr{A}$, searching for skew-symmetric binary
sequences with odd length $n$ and high merit factor, to an algorithm
$\mathscr{B}$, searching simultaneously skew-symmetric binary sequences with
odd length $n$ and pseudo-skew-symmetric binary sequences with even lengths
$n-1$ and $n+1$.

##  IV Restricted Skew-Symmetric Sequences Classes

Throughout the rest of this paper, we use the following definition of
restriction.

######  Definition 2 (Restriction Class of Binary Sequence)

We will call the class of binary sequences of length $n$, with the first $k$
elements fixed, a restriction class of order $k$ on binary sequences with
length $n$. We will denote this set as $R_{n}^{k}$. If the binary sequence is
skew-symmetric we will use the notation $\mathscr{R}_{n}^{k}$.

It should be noted that $\mathscr{R}_{n}^{k}\subset R_{n}^{k}$. More
precisely, the magnitude of $R_{n}^{k}$ is $2^{n-k}$, while the magnitude of
$\mathscr{R}_{n}^{k}$ is $2^{l-k+1}$, where $n=2l+1$, since
$\mathscr{R}_{n}^{k}$ is defined over the skew-symmetric binary sequences
only.

A well-studied area in number theory and combinatorics is the number partition
problem - distinct ways of writing a given integer number $n$ as a sum of
positive integers. We define the number of possible partitions of a non-
negative integer $n$ as the partition function $p(n)$. No closed-form
expression for $p(n)$ is known. However, the partition functions for some
different values of $n$ could be found in the online encyclopedia of integer
numbers (OEIS), sequence A000041 [23].

Theoretically, searching for skew-symmetric binary sequences of length $n$
with high merit factors could be parallelized to $\mid\mathscr{R}_{n}^{k}\mid$
instances. To minimize the total number of instances needed, we should
consider several actions to a given skew-symmetric binary sequence
$B=(b_{0},b_{1},\cdots,b_{n-1})$:

  * •

Reversing $B$ defined as operator $\delta_{1}$:
$\delta_{1}(B)=(b_{n-1},\cdots,b_{1},b_{0})$

  * •

Complementing $B$ defined as operator $\delta_{2}$:
$\delta_{2}(B)=(\overline{b_{0}},\overline{b_{1}},\cdots,\overline{b_{n-1}})$,
where $\overline{b_{i}}=-b_{i}$

  * •

Alternating complementing of $B$ defined as operator $\delta_{3}$ :
$\delta_{3}(B)=(\cdots,\overline{b_{i-2}},b_{i-1},\overline{b_{i}},b_{i+1},\overline{b_{i+2}},\cdots)$

All three operators leaves the energy of $B$ intact. If we further add the
identity operator $\delta_{0}$ we construct a group $G$ of order 8. By using
some group theory [5], we could derive a closed formula of the exact number of
symmetry classes with length $k$:
$2^{k-3}+2^{\lfloor\frac{k}{2}\rfloor-2+\left(k\text{ mod }2\right)}$. The
same formula arises from the row sums of the Losanitsch’s triangle (OEIS,
sequence A005418 [24]) - named after the S. Lozanić, in his work related to
the symmetries exhibited by rows of paraffins [25]. This fact could be used to
partition the search space from $p(k)$ covering subsets to
$2^{k-3}+2^{\lfloor\frac{k}{2}\rfloor-2+\left(k\text{ mod }2\right)}$ non-
covering subsets. A similar partitioning was used in [5] to efficiently
parallelize a branch and bound algorithm for exhaustively search binary
sequences with optimal merit factors. Since exhaustive search is inapplicable
for large values of $n$, the following characteristic is proposed:

######  Definition 3 (Potential of a Restriction Subclass)

For a skew-symemtric binary sequence $B=(b_{0},b_{1},\cdots,b_{n-1})$, we fix
a partitioning with length $k$: $t_{0},t_{1},\cdots,t_{g}$, s.t.
$\sum_{i=0}^{g}{t_{i}}=k$. The partitioning could be projected to a skew-
symmetric binary sequence with the following procedure:

$R=\underbrace{a\cdots
a}_{t_{0}}\underbrace{\overline{a}\cdots\overline{a}}_{t_{1}}\underbrace{a\cdots
a}_{t_{2}}\underbrace{\overline{a}\cdots\overline{a}}_{t_{3}}\cdots\underbrace{(-1)^{g}a\cdots(-1)^{g}a}_{t_{g}}\underbrace{u_{1}u_{2}u_{3}\cdots
u_{n-2k-2}u_{n-2k-1}u_{n-2k}}_{\text{ non-fixed (free)
elements}}\underbrace{f_{1}f_{2}f_{3}\cdots f_{k-2}f_{k-1}f_{k}}_{\text{ last
elements are fixed}}$

Last $k$ elements $f_{i}$ are fixed due to the first $k$ elements of the
sequence and its skew-symmetric property. Please note that all elements
$a,\overline{a},(-1)^{g}a,u_{i},f_{i}\in\left\\{-1,1\right\\}$. We define the
potential of the binary skew-symmetric sequence $R$ as the energy of the
ternary sequence $R^{z}$, where:

$R^{z}=\underbrace{a\cdots
a}_{t_{0}}\underbrace{\overline{a}\cdots\overline{a}}_{t_{1}}\underbrace{a\cdots
a}_{t_{2}}\underbrace{\overline{a}\cdots\overline{a}}_{t_{3}}\cdots\underbrace{(-1)^{g}a\cdots(-1)^{g}a}_{t_{g}}\underbrace{000\cdots
000}_{n-2k\text{ zeroed elements}}\underbrace{f_{1}f_{2}f_{3}\cdots
f_{k-2}f_{k-1}f_{k}}_{\text{ last elements are fixed}}$

$R^{z}$ is ternary since we have introduced a new element $0$. This way we
could not only focus on the complete sidelobes of $R$ but take under
consideration the non-complete fragments of sidelobes of $R$, where the fixed
elements of the sequence play a role. For example, let us consider a skew-
symmetric binary sequence $Q$ with length $n=21$, a restriction $k=6$ and a
partition $1,1,2,2$:

$Q=\underbrace{a}_{1}\underbrace{\overline{a}}_{1}\underbrace{aa}_{2}\underbrace{\overline{a}\overline{a}}_{2}\underbrace{u_{1}u_{2}u_{3}\cdots
u_{9}}_{\text{ non-fixed (free)
elements}}\underbrace{f_{1}f_{2}f_{3}f_{4}f_{5}f_{6}}_{\text{elements are
fixed}}$

Since $Q$ is skew-symmetric we know that $Q[l-i]=(-1)^{i}Q[l+i]$, for
$n=2l+1$. If we take $i=l$ we have $Q[0]=(-1)^{l}Q[n-1]$. In the current
example, $n=21$ and $l=10$. Therefore $f_{6}=Q[20]=Q[0](-1)^{l}=Q[0]$. By
following the same routine we could reveal all values of $f_{i}$:

$Q=\underbrace{a}_{1}\underbrace{\overline{a}}_{1}\underbrace{aa}_{2}\underbrace{\overline{a}\overline{a}}_{2}\underbrace{u_{1}u_{2}u_{3}\cdots
u_{9}}_{\text{ non-fixed (free)
elements}}\underbrace{a}_{1}\underbrace{\overline{a}\overline{a}}_{2}\underbrace{aaa}_{3}$

We could easily derive $Q^{z}$:

$Q^{z}=\underbrace{a}_{1}\underbrace{\overline{a}}_{1}\underbrace{aa}_{2}\underbrace{\overline{a}\overline{a}}_{2}\underbrace{000000000}_{9}\underbrace{a}_{1}\underbrace{\overline{a}\overline{a}}_{2}\underbrace{aaa}_{3}$

Without loss of generality, let us fix $a=1$. Then, we have $\overline{a}=-1$,
and

$Q^{z}=(1,-1,1,1,-1,-1,0,0,0,0,0,0,0,0,0,1,-1,-1,1,1,1)$

Thus, the potential of the partition $1,1,2,2$ is equal to
$\mathbb{E}(Q^{z})$. The sidelobes’ array of $Q^{z}$ is

$S_{Q^{z}}=\left[1,0,1,0,1,0,-5,0,3,0,-1,0,0,0,0,0,0,0,-4,0\right],$

therefore $\mathbb{E}(Q^{z})=\sum_{u}{S_{Q^{z}}}_{u}^{2}=54.$ The cardinality
of the set $\mathscr{R}_{21}^{6}$ is
$\mid\mathscr{R}_{21}^{6}\mid=2^{6-3}+2^{\lfloor\frac{6}{2}\rfloor-2+\left(6\text{
mod }2\right)}=2^{3}+2^{1}=10$. A list of unique partitions in
$\mathscr{R}_{21}^{6}$ could be find in Table I. For simplicity, we denote as
$\mathscr{R}_{n}^{k\mid m}$ those partitions of size $k$ over $n$, which
posses exactly $m$ elements. For example, referring Table I, the partition
$(6)$ is in $\mathscr{R}_{21}^{6\mid 1}$, partitions $(5,1)$, $(4,2)$ and
$(3,3)$ are in $\mathscr{R}_{21}^{6\mid 2}$, partitions $(4,1,1)$, $(3,1,2)$,
$(3,2,1)$ and $(2,2,2)$ are in $\mathscr{R}_{21}^{6\mid 3}$, while partitions
$(2,1,2,1)$ and $(2,1,1,2)$ are in $\mathscr{R}_{21}^{6\mid 4}$.

TABLE I: A list of unique partitions in $\mathbb{R}_{21}^{6}$ Partition |  $\pm$ Notation
---|---
$6$ | [’+’, ’+’, ’+’, ’+’, ’+’, ’+’]
$5,1$ | [’+’, ’+’, ’+’, ’+’, ’+’, ’-’]
$4,1,1$ | [’+’, ’+’, ’+’, ’+’, ’-’, ’+’]
$4,2$ | [’+’, ’+’, ’+’, ’+’, ’-’, ’-’]
$3,1,2$ | [’+’, ’+’, ’+’, ’-’, ’+’, ’+’]
$3,2,1$ | [’+’, ’+’, ’+’, ’-’, ’-’, ’+’]
$3,3$ | [’+’, ’+’, ’+’, ’-’, ’-’, ’-’]
$2,1,2,1$ | [’+’, ’+’, ’-’, ’+’, ’+’, ’-’]
$2,1,1,2$ | [’+’, ’+’, ’-’, ’+’, ’-’, ’-’]
$2,2,2$ | [’+’, ’+’, ’-’, ’+’, ’-’, ’-’]

Given a partition $t_{0},t_{1},\cdots,t_{g}$ of size $k$, we will denote the
set of skew-symmetric binary sequences defined by the partition as
$\mathbb{B}_{n}^{t_{0},t_{1},\cdots,t_{g}}$. Please note that
$\mathbb{B}_{n}^{t_{0},t_{1},\cdots,t_{g}}\subset\mathscr{R}_{n}^{k\mid
g+1}\subset\mathscr{R}_{n}^{k}$. Finally, the potential of a given partition
set $S$ is denoted as $\mathscr{U}(S)$.

Few remarks regarding the sidelobes of a given potential ternary sequence
should be made. In case we are interested in the potential of
$\mathscr{R}_{n}^{k}$, the sidelobes of the ternary sequence could be divided
into three distinct sections:

  * •

Head: first $2k$ sidelobes. The first $k$ sidelobes are shared among all
sequences in this class, i.e. they are immutable. The next $k$ sidelobes are
just partially revealed.

  * •

Body: the mid $n-3k$ sidelobes. They are all equal to zero.

  * •

Tail: the last $k$ sidelobes. Ignoring the sidelobes equal to zero, the
remaining sidelobes are even numbers greater than 0. They are not shared among
the sequences in this class, i.e. they are mutable. However, partial
information about their final value is gathered.

The actual calculation of the potential $\mathscr{U}(\mathscr{R}_{n}^{k})$
gives an equal priority to the value of the elements in the head and the tail.
However, we could tweak the actual energy calculation while minimizing the
energy of the elements to prefer minimizing the elements in the head more,
than minimizing the elements in the tail. All elements inside the tail are
even numbers. If we prefer to minimize the energy of the summands, rather than
minimizing their overall sums, we could normalize the tail by dividing its
sidelobes values by 2. We will define this value as a normalized potential,
denoted as $\mathscr{U}^{\star}(\mathscr{R}_{n}^{k})$.

As a final remark, please note that despite
$\mathscr{R}_{n}^{k},\mathscr{R}_{n+1}^{k},\mathscr{R}_{n+2}^{k},\cdots$ is an
infinite sequence of non-intersecting finite sets, their potentials and
normalized potentials are equal. More formally, $\forall i\geq n\forall j\geq
i:\mathscr{U}(\mathscr{R}_{i}^{k})=\mathscr{U}(\mathscr{R}_{j}^{k})\&\mathscr{U}^{\star}(\mathscr{R}_{i}^{k})=\mathscr{U}^{\star}(\mathscr{R}_{j}^{k})$.

During our research, by using an exhaustive search, we have calculated all the
potentials, as well as normalized potentials, of set partitions of the form
$\mathscr{R}_{n}^{k\mid g}$, for $38<k<115$ and some values of
$g\in\left[4,12\right]$. For speeding up the exhaustive routine, the following
restriction of the partitions were further applied: $\forall i:t_{i}\geq
t_{i+1}$. As an illustration, various partitions having an optimal potential
and normalized potential are given in Table II.

TABLE II: Some partitions with optimal and normalized potentials Class |  $\mathscr{U}$ optimal | $\mathscr{U}$ |  $\mathscr{U}^{\star}$ optimal | $\mathscr{U}^{\star}$
---|---|---|---|---
$\mathscr{R}_{n}^{39\mid 4}$ | 18,11,6,4 | 3731 | 18,11,6,4 | 1082
$\mathscr{R}_{n}^{41\mid 6}$ | 17,9,6,4,3,2 | 2217 | 17,9,6,4,3,2 | 813
$\mathscr{R}_{n}^{47\mid 9}$ | 18,8,5,4,3,3,2,2,2 | 1859 | 11,9,5,5,5,3,3,3,3 | 830
$\mathscr{R}_{n}^{56\mid 4}$ | 27,14,9,6 | 12856 | 27,14,9,6 | 3472
$\mathscr{R}_{n}^{68\mid 7}$ | 25,11,10,7,5,5,5 | 9596 | 25,12,9,8,6,4,4 | 3040
$\mathscr{R}_{n}^{79\mid 9}$ | 26,12,10,7,6,6,6,4,2 | 11667 | 28,14,10,7,6,6,4,2,2 | 3702

##  V Algorithm for Finding Binary Sequences with Arbitrary Length and High
Merit Factor

In our previous work, we were able to reduce the memory complexity of state-
of-the-art algorithms from $O\left(n^{2}\right)$ to $O\left(n\right)$, by
keeping the time complexity linear as well. This was achieved by the usage of
the following theorem:

###### Theorem 1

Given two skew-symmetric sequences $B$ and $B^{q}$ with length $n=2l+1$, where
$B^{q}$ corresponds to $B$ with $q$-th and $n-q-1$-th bit flipped for some
fixed $q<l$, and with sidelobe array of $B$ denoted as $S$, the following
property holds:

$\begin{split}&\mathbb{E}(B^{q})=\mathbb{E}(B)+\sum_{r=q+1,r\neq
2q+1}^{n-q-1}(16+\sigma\kappa\epsilon_{1})+\sum_{r=n-q,r\neq
2q+1}^{n-1}(\kappa(\epsilon_{2}+\sigma\epsilon_{1})+32+32\sigma\epsilon_{1}\epsilon_{2})+\sum_{r\geq
n-q,r\leq n-1,r=2q+1}(16+\kappa\epsilon_{2}),\end{split}$ (5)

where $\sigma=(-1)^{l-q}$, $\kappa=-8S_{r}L[q]$, $\epsilon_{1}(r)=B[r-q-1]$,
$\epsilon_{2}(r)=B[q+r-n]$.

The proof of Theorem 1 could be found in [26].

By achieving both linear time and memory complexities, we can utilize all the
threads of a given central processing unit. Furthermore, the memory
requirements of a given algorithm are significantly reduced.

Algorithm 1 Algorithm for searching skew-symmetric and pseudo-skew-symmetric
binary sequences with arbitrary lengths and high merit factors.

1:procedure
MF($n,t_{0},t_{1},\cdots,t_{g},\mathbb{T}_{i},\mathbb{T}_{o},\mathbb{T}_{a}$)

2: $\bot_{n-1}$, $\bot_{n}$, $\bot_{n+1}$, $w_{o}\leftarrow 0,0,0,0$

3: while True do

4: $\mathbb{H},w_{i},\leftarrow\left\\{\emptyset\right\\},0$

5: $B\leftarrow$ random , s.t.
$B\in\mathbb{B}_{n}^{t_{0},t_{1},\cdots,t_{g}}\subset\mathscr{R}_{n}^{k},$ for
$k=\sum_{i=0}^{g}{t_{i}}$.

6: $\mathbb{H}$.add(hash$(B)$)

7: $V\leftarrow\mathbb{E}(B)$

8: while True do

9: bestN $\leftarrow$ pickBetterNeighborIndex$(B)$

10: if bestN $==-1$ then

11: break

12: end if

13: Flip(bestN, $B$)

14: $V\leftarrow\mathbb{E}(B)$

15: $w_{i}\mathrel{+}=1$

16: $\mathbb{H}$.add(hash($B$))

17: if $\frac{n^{2}}{2V}>\bot_{n}$ then

18: $\bot_{n}\leftarrow\frac{n^{2}}{2V}$

19: end if

20: if $\frac{n^{2}}{2V}\geq\mathbb{T}_{a}$ then

21: if $\frac{{(n+1)}^{2}}{2(V+n+2b_{n}\delta)}>\bot_{n+1}$ then

22: $\bot_{n+1}\leftarrow\frac{{(n+1)}^{2}}{2(V+n+2b_{n}\delta)}$

23: end if

24: if $\frac{{(n-1)}^{2}}{2(V+n-3+2b_{n-1}\delta)}>\bot_{n-1}$ then

25: $\bot_{n-1}\leftarrow\frac{{(n-1)}^{2}}{2(V+n-3+2b_{n-1}\delta)}$

26: end if

27: end if

28: if $w_{i}>\mathbb{T}_{i}$ then

29: $w_{o}\mathrel{+}=1$

30: break

31: end if

32: end while

33: if $w_{o}>\mathbb{T}_{o}$ then

34: break

35: end if

36: end while

37:end procedure

In Algorithm 1 a pseudo-code of the proposed routine is presented. The
following additional notations and remarks should be considered:

  * •

$n$ \- an odd integer number

  * •

$t_{0},t_{1},\cdots,t_{g}$ \- the partition search space to search through.

  * •

$\mathbb{T}_{i}$ \- an inner threshold value. When the inner counter $w_{i}$
reaches $\mathbb{T}_{i}$, the set is flushed and the whole routine restarts.
The threshold value $\mathbb{T}_{i}$ constrains the size of the set
$\mathbb{H}$.

  * •

$\mathbb{T}_{o}$ \- an outer threshold value. When the outer counter $w_{o}$
reaches $\mathbb{T}_{o}$, the program is terminated.

  * •

$\mathbb{T}_{a}$ \- an activator threshold value. For example, the probability
of finding pseudo-skew-symmetric sequence with length $n-1$ or $n+1$ and merit
factor $X$, from a skew-symmetric sequence with length $n$ and merit factor
$X-1$, is negligible for higher values of $X$. Thus, we could save time and
efforts to repeatedly probe the adjacent pseudo-skew-symmetric sequences.

  * •

$\bot_{n-1},\bot_{n},\bot_{n+1}$ \- the best candidates found, in terms of
merit factor value, for respectively pseudo and not pseudo-skew-symmetric
sequences of lengths $n-1,n$ and $n+1$.

  * •

$\mathbb{H}$ \- a set of hashes of the visited candidates. We make sure to
avoid already visited nodes.

  * •

$\mathbb{H}$.add(hash$(B)$) - adding the hash of the binary sequence $B$ to
the set $\mathbb{H}$.

  * •

pickBetterNeighborIndex - a function, which returns the index of a better
unexplored neighbor of $B$, i.e. the binary sequence with a distance of
exactly 1 flip away from $B$, s.t. its hash does not belong to the set
$\mathbb{H}$. An optimized derivative-based pseudo-code of this helper
function is discussed in our previous work [26].

##  VI Results

Algorithm 1 was implemented (C++) on a general-purpose computer equipped with
a central processing unit with 8 cores and 16 threads. Despite using just a
single low-budget personal computer, we were able to improve all the results,
for all skew-symmetric lengths in the range 225-451, announced in literature
and reached by using a supercomputer grid. Furthermore, by using classes of
pseudo-skew-symmetric sequences, we were able to simultaneously reach binary
sequences of even lengths between 225 and 512, and beyond, with merit factors
greater than 7. We demonstrate the efficiency of our approach by publishing a
complete list of binary sequences, for both even and odd lengths up to
$2^{8}$, and merit factors greater than 8. The list is further accompanied by
a complete list of binary sequences, for both even and odd lengths up to
$2^{9}$, and merit factors greater than 7 (see Tables IV \- XIV).

We further demonstrate the power and efficiency of the proposed algorithm by
launching it on binary sequences of lengths 573 and 1009. As mentioned
earlier, the choice of those two specific lengths is motivated by the
approximation numbers given in [15], Figure 7, presented during a discussion
of how much time the state-of-the-art stochastic solver lssOrel_8 will need to
reach binary sequences with the aforementioned lengths and merit factors close
to $6.34$. It was estimated that finding solutions with a merit factor of 6.34
for a binary sequence with length 573 requires around 32 years, while for
binary sequences with length 1009, the average runtime prediction to reach the
merit factor of 6.34 is $46774481153$ years. By using the proposed in this
work algorithm, we were able to reach such candidates within several hours
(see Table XV). By further applying some operators on the skew-symmetric
binary sequence of length 1009 found, several sequences of lengths 1006, 1007,
1008, and 1010 with MF greater than 6.34 were also revealed. The same argument
is true for the other sequence of length 573, but since the results are too
many we omit the data.

For convenience, we denote the operators acting on binary sequences as shown
in Table III. Please note that operator $\eta_{0}$ activated on a given skew-
symmetric binary sequence $a||L||b$ will yield another skew-symmetric binary
sequence $L$, while all other operators activated on the same skew-symmetric
binary sequence will yield a pseudo-skew-symmetric sequence. Throughout the
tables with reported records, the classes denoted with $\Omega$ represent the
best-known result to be found in the literature for the current length (all in
Table IV, for the lengths between 172 and 226). It should be emphasized, that
all records achieved by starting from a sequence of class $\Omega$, are
directly calculated without the usage of any additional stochastic routine.
All other records throughout the tables (classes $\mathbb{B}$) are achieved by
using a heuristic search. All sequences are presented in hexadecimal format
with zeroes omitted. It should be noted, that as soon as the algorithm finds a
record sequence of a given length, it automatically continues to the next
search space. In some cases, we required a little bit more demanding goal,
i.e. MF greater than 8 (for sequences with lengths less than about 256), or MF
greater than 7 (for sequences with lengths less than about 512). Some records
were found for several minutes, while others required a little bit more effort
of several hours.

TABLE III: A list of used operators acting on binary sequences Operator | Action
---|---
$\eta_{0}$ | $a||L||b\circ\eta_{0}=L$
$\eta_{1}$ | $L\circ\eta_{1}=L||1$
$\eta_{2}$ | $L\circ\eta_{2}=L||-1$
$\eta_{3}$ | $a||L\circ\eta_{3}=L$
$\eta_{4}$ | $L||b\circ\eta_{4}=L$
$\eta_{5}$ | $L\circ\eta_{5}=1||L$
$\eta_{6}$ | $L\circ\eta_{6}=-1||L$

##  VII Conclusions

In this work, several new sieving classes of binary sequences with high merit
factors are proposed. A class of finite binary sequences of even length called
pseudo-skew-symmetric with alternate absolute values equal to 1 is found. We
show how algorithms searching for skew-symmetric binary sequences with high MF
of odd length, could be easily transformed to search pseudo-skew-symmetric
binary sequences of even length. We have further introduced the concept of
potentials measured by number partitioning and helper ternary sequences. This
allows us to reach binary sequences of any length with record merit factors in
record times. Numerical pieces of evidence suggest that there are various non-
intersecting classes of binary sequences with asymptotic values greater than
7, and even 8, which is significantly greater than the current largest known
asymptotic merit factor of approximately 6.34.

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TABLE IV: A list of binary sequences with record merit factor values $n$ | Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$172$ | $\Omega_{173}\circ\eta_{4}$ |  fe03184fe780309206b6663e571d355a6ac59356a | 8.8363 | 9.052631578947368
$178$ | $\Omega_{179}\circ\eta_{4}$ |  2c7bd3a7034ccbfe886e8a084688550ccf2b613d16c | 8.2125 | 9.29149560117302
$180$ | $\Omega_{179}\circ\eta_{1}$ |  b1ef4e9c0d332ffa21ba28211a2154333cad84f45b0 | 8.5353 | 8.553326293558607
$182$ | $\Omega_{183}\circ\eta_{4}$ |  3cfe712191dc7d1c57c81ec5a8d7edbd6ddb9a3b654d2 | 8.2194 | 8.928301886792452
$184$ | $\Omega_{183}\circ\eta_{2}$ |  f3f9c4864771f4715f207b16a35fb6f5b76e68ed9534a | 8.2980 | 8.636734693877552
$186$ | $\Omega_{185}\circ\eta_{1}$ |  233da5ef19ed00149bc0c4644d4b9c1550e992e878b375 | 8.3606 | 8.627431421446383
$190$ | $\Omega_{187}\circ\eta_{0}\circ\eta_{0}\circ\eta_{4}$ |  190c8d2692191f17dba56aff75407e11d7b5b9a3863c8cb9 | 8.5021 | 9.195109526235354
$192$ | $\Omega_{195}\circ\eta_{4}\circ\eta_{4}\circ\eta_{4}$ |  190c8d2692191f17dba56aff75407e11d7b5b9a3863c8cb9 | 8.0000 | 8.930232558139535
$193$ | $\Omega_{195}\circ\eta_{4}\circ\eta_{4}$ |  32191a4d24323e2fb74ad5feea80fc23af6b73470c791973 | 8.6868 | 9.11179060665362
$193$ | $\Omega_{195}\circ\eta_{0}\circ\eta_{0}\circ\eta_{1}\circ\eta_{1}$ |  c864693490c8f8bedd2b57fbaa03f08ebdadcd1c31e465cf | 8.6868 | 9.238343253968255
$194$ | $\Omega_{195}\circ\eta_{0}\circ\eta_{0}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}$ |  190c8d2692191f17dba56aff75407e11d7b5b9a3863c8cb9f | 8.0522 | 8.644005512172715
$196$ | $\Omega_{199}\circ\eta_{4}\circ\eta_{4}\circ\eta_{4}$ |  937e64c9f4bc13e78367a16729653ad0f58ce65738aaa | 8.0910 | 8.521739130434783
$198$ | $\Omega_{199}\circ\eta_{4}$ |  24df99327d2f04f9e0d9e859ca594eb43d633995ce2aaa | 8.3662 | 8.786194531600179
$200$ | $\Omega_{199}\circ\eta_{2}$ |  937e64c9f4bc13e78367a16729653ad0f58ce65738aaaa | 8.2919 | 8.756567425569177
$202$ | $\Omega_{199}\circ\eta_{2}\circ\eta_{1}\circ\eta_{6}$ |  126fcc993e97827cf06cf42ce52ca75a1eb19ccae715555 | 8.0291 | 8.568668626627467
$204$ | $\Omega_{205}\circ\eta_{4}$ |  3c7877d72fc246a45d9aaedfb63a8eff98847e474af20a25a4b | 8.1858 | 8.276849642004773
$206$ | $\Omega_{207}\circ\eta_{4}$ |  2492010c9e4ed276a103f76a13a07752ba0763c4e58cbaa38e2 | 7.7921 | 8.436580516898609
$208$ | $\Omega_{211}\circ\eta_{0}\circ\eta_{4}$ |  38e383be228cf9981fc150c0fafad4d014a9599acdf76bb5b6db | 7.9529 | 8.96849087893864
$209$ | $\Omega_{211}\circ\eta_{4}\circ\eta_{4}$ |  38e383be228cf9981fc150c0fafad4d014a9599acdf76bb5b6db | 8.6394 | 8.849473257698541
$209$ | $\Omega_{211}\circ\eta_{0}$ |  71c7077c4519f3303f82a181f5f5a9a02952b3359beed76b6db6 | 8.6394 | 9.254449152542373
$210$ | $\Omega_{211}\circ\eta_{4}$ |  71c7077c4519f3303f82a181f5f5a9a02952b3359beed76b6db6 | 7.9862 | 9.153175591531756
$212$ | $\Omega_{213}\circ\eta_{4}$ |  2a2fbaa406cf5693c8d89b658f12d8639d8dcb1e02ce547fbaf7f | 7.8082 | 9.262984336356142
$214$ | $\Omega_{213}\circ\eta_{1}$ |  a8beea901b3d5a4f23626d963c4b618e76372c780b3951feebdfd | 8.1808 | 9.17755511022044
$216$ | $\Omega_{215}\circ\eta_{1}$ |  7c361f6410df47fc0c506cc111bbacc6becad56f5cbae75a72d6b | 7.8598 | 8.589101620029455
$218$ | $\Omega_{217}\circ\eta_{1}$ |  1a403f08432daddf13836660a67d66635b1288f8f345d2b547955 | 7.4888 | 8.328776726253066
$220$ | $\Omega_{221}\circ\eta_{4}$ |  3f06f260dece7d91c596a6d0009d550e7e184918a6ce8d672e52b 5 | 7.2848 | 8.479327259985984
$222$ | $\Omega_{223}\circ\eta_{4}$ |  1f1f0fa3be027fcc798db8f201889aa3491c996cd562a51214b5b 5a | 7.6742 | 8.71666077113548
$224$ | $\Omega_{223}\circ\eta_{2}$ |  7c7c3e8ef809ff31e636e3c806226a8d247265b3558a944852d6d 6a | 7.3962 | 8.521739130434783
$226$ | $\Omega_{225}\circ\eta_{1}$ |  e060603dfef3807b8dce68f58e36d82de6c8dba55b2ea8b565656 d5 | 7.1555 | 8.487205051512131
$227$ | $\mathbb{B}_{n}^{15,11,7}$ |  7fff001fc3cdfb67e939a58cc0d8658d4cd879b1ea63a8cb4a955 2aaa | 7.5578 | 8.069057312871907
$228$ | $\mathbb{B}_{228}^{17}$ |  ffff9c31639d28ccf5ab85cba46d3c6e10de901f4cc83d927b2d9 5555 | 7.2888 | 8.18903591682
$229$ | $\mathbb{B}_{228}\circ\eta_{5}$ |  1ffff9c31639d28ccf5ab85cba46d3c6e10de901f4cc83d927b2d 95555 | 7.5476 | 8.55202217873
$230$ | $\mathbb{B}_{228}\circ\eta_{1}\circ\eta_{5}$ |  3ffff3862c73a5199eb570b9748da78dc21bd203e99907b24f65b 2aaab | 7.1739 | 8.16610064835
$231$ | $\mathbb{B}_{231}^{16}$ |  7fffb612c7d8bc368ed13b8379234c371a35bb10ede34bd8a4f16 3aaaa | 7.4381 | 8.18671371586
$232$ | $\mathbb{B}_{233}\circ\eta_{4}$ |  fffad0296b1ca397ca63d8cedc5b991edc4c9d260d7920db0782b c1555 | 7.1727 | 8.05748502994
$233$ | $\mathbb{B}_{233}^{13}$ |  1fff5a052d639472f94c7b19db8b7323db8993a4c1af241b60f05 782aaa | 7.3522 | 8.22560606061
$234$ | $\mathbb{B}_{233}\circ\eta_{5}$ |  3fff5a052d639472f94c7b19db8b7323db8993a4c1af241b60f05 782aaa | 7.1651 | 8.27379873073
$235$ | $\mathbb{B}_{233}^{15}\circ\eta_{5}\circ\eta_{5}$ |  7fff5a052d639472f94c7b19db8b7323db8993a4c1af241b60f05 782aaa | 7.3496 | 8.44677271337
$236$ | $\mathbb{B}_{233}\circ\eta_{1}\circ\eta_{5}\circ\eta_{5}$ |  fffeb40a5ac728e5f298f633b716e647b7132749835e4836c1e0a f05555 | 7.5797 | 8.37282020445
$237$ | $\mathbb{B}_{239}\circ\eta_{0}$ |  1fff3863ff634e14a46fe933d8c162c27ac9d338546e0fa4f2755 2693555 | 7.8230 | 8.65203327172
$238$ | $\mathbb{B}_{239}^{14}\circ\eta_{4}$ |  3fff3863ff634e14a46fe933d8c162c27ac9d338546e0fa4f2755 2693555 | 7.7573 | 8.34226804124
$239$ | $\mathbb{B}_{239}^{19}$ |  7ffff0c39e1f03f01ef8967c933666e66331ca61dae952b52969b 4d2aaaa | 7.6962 | 8.19056495555
$240$ | $\mathbb{B}_{239}^{14}\circ\eta_{2}$ |  fffcff800f3c398721e6e43326f40dcaf46332e465a36992d34aa 954d554 | 7.2948 | 8.24742268041
$241$ | $\mathbb{B}_{241}$ |  1ffffc4235e31e3c1ee6079bc2973b0b13782d196a645ad25b25f 22ed5555 | 8.0668 | 8.26893507973
$242$ | $\mathbb{B}_{241}\circ\eta_{2}$ |  3ffff8846bc63c783dcc0f37852e761626f05a32d4c8b5a4b64be 45daaaaa | 7.1893 | 8.14973559699
$243$ | $\mathbb{B}_{243}^{18}$ |  7ffff8c0e7381f840fe3960f3a4c66ce64c7b2d61b6ad45a95b26 d4daaaaa | 7.2488 | 8.46216680997

TABLE V: A list of binary sequences with record merit factor values $n$ | Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$244$ | $\mathbb{B}_{243}\circ\eta_{1}$ |  fffff181ce703f081fc72c1e7498cd9cc98f65ac36d5a8b52b64d a9b55555 | 7.1730 | 8.51974813967
$245$ | $\mathbb{B}_{243}\circ\eta_{1}\circ\eta_{5}$ |  1fffff181ce703f081fc72c1e7498cd9cc98f65ac36d5a8b52b64 da9b55555 | 7.3237 | 8.5799028016
$246$ | $\mathbb{B}_{243}\circ\eta_{1}\circ\eta_{1}\circ\eta_{5}$ |  3ffffe3039ce07e103f8e583ce9319b39931ecb586dab516a56c9 b536aaaab | 7.1650 | 8.33324153126
$247$ | $\mathbb{B}_{243}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}$ |  7ffffc60739c0fc207f1cb079d2633673263d96b0db56a2d4ad93 6a6d55556 | 7.109 | 8.22889128675
$248$ | $\mathbb{B}_{243}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}\circ\eta_{6}$ |  7ffffc60739c0fc207f1cb079d2633673263d96b0db56a2d4ad93 6a6d55556 | - | 8.01668404588
$249$ | $\mathbb{B}_{249}^{16}$ |  1ffffc212fe40e94e19e33d2972665b1b1e663783d3259a4f84ae 543a2d5555 | 8.1323 | 8.20119047619
$250$ | $\mathbb{B}_{249}\circ\eta_{2}$ |  3ffff8425fc81d29c33c67a52e4ccb6363ccc6f07a64b349f095c a8745aaaaa | 7.1988 | 8.19564647259
$251$ | $\mathbb{B}_{249}\circ\eta_{2}\circ\eta_{1}$ |  7ffff084bf903a538678cf4a5c9996c6c7998de0f4c96693e12b9 50e8b55555 | 7.5632 | 8.2354248366
$252$ | $\mathbb{B}_{253}\circ\eta_{4}$ |  ffffe03198f80ce61e0f2cf07a25ce2176c877a52cf2d6966cd5a d99356aaaa | 7.2394 | 8.49438202247
$253$ | $\mathbb{B}_{253}^{19}$ |  1ffffc06331f019cc3c1e59e0f44b9c42ed90ef4a59e5ad2cd9ab 5b326ad5555 | 7.3036 | 8.95481253497
$254$ | $\mathbb{B}_{253}\circ\eta_{2}$ |  3ffff80c663e03398783cb3c1e8973885db21de94b3cb5a59b356 b664d5aaaaa | 7.0325 | 8.51808819646
$255$ | $\mathbb{B}_{253}\circ\eta_{2}\circ\eta_{2}$ |  7ffff018cc7c06730f0796783d12e710bb643bd296796b4b366ad 6cc9ab55554 | 7.2849 | 8.22892938497
$256$ | $\mathbb{B}_{259}\circ\eta_{0}\circ\eta_{4}$ |  fff65cbd0a16c7841bd24913277f2064c6572a27311c70b945a4e 17d0bc862aa | 7.1483 | 8.12698412698
$257$ | $\mathbb{B}_{259}\circ\eta_{0}$ |  1ffffc380df2781e7233e42db41b66c64e6c671af1c2e532365a9 635ca92d5555 | 7.3847 | 8.32270665323
$258$ | $\mathbb{B}_{259}\circ\eta_{4}$ |  3ffffc380df2781e7233e42db41b66c64e6c671af1c2e532365a9 635ca92d5555 | 7.0738 | 8.25241755517
$259$ | $\mathbb{B}_{259}^{19}$ |  7ffff8701be4f03ce467c85b6836cd8c9cd8ce35e385ca646cb52 c6b9525aaaaa | 8.0918 | 8.18659995118
$260$ | $\mathbb{B}_{259}^{16}\circ\eta_{1}$ |  ffff3e0648fcf2598f931e43a9538a717b6093f812e5b39499e34 d48e6a53555 | - | 8.26405867971
$261$ | $\mathbb{B}_{261}$ |  1fffff21274a5ec18d9e601cf948f139c2d93b48f94daa659c9ac 5e0f63a35555 | - | 8.04452054795
$262$ | $\mathbb{B}_{263}^{16}\circ\eta_{4}$ |  3ffff3d98cc67b60b077887e1c1eed486c68fc45ada568976b0a7 166cc99d3555 | 7.0231 | 8.08146927243
$263$ | $\mathbb{B}_{263}^{16}$ |  7fffe7b3198cf6c160ef10fc383dda90d8d1f88b5b4ad12ed614e 2cd9933a6aaa | 7.2006 | 8.22852724245
$264$ | $\mathbb{B}_{265}\circ\eta_{4}$ |  ffffec3731980ffc25863ed306f12b6b503e3f12e530eb65874aa d5993234eaaa | 7.1235 | 8.17260787993
$265$ | $\mathbb{B}_{267}\circ\eta_{0}$ |  1ffffe3943ca79588ed4f2ccc37e13e24dce253a572ccc34fc489 f960d2f925555 | 7.0963 | 8.71710526316
$266$ | $\mathbb{B}_{265}\circ\eta_{1}$ |  3ffffb0dcc6603ff09618fb4c1bc4adad40f8fc4b94c3ad961d2a b5664c8d3aaaa | - | 8.10492554410
$267$ | $\mathbb{B}_{267}^{20}$ |  7ffffc728794f2b11da9e59986fc27c49b9c4a74ae599869f8913 f2c1a5f24aaaa | 7.0765 | 8.07715839565
$268$ | $\mathbb{B}_{267}\circ\eta_{2}\circ\eta_{2}\circ\eta_{3}$ |  fffff1ca1e53cac476a796661bf09f126e7129d2b96661a7e244f cb0697c92aaaa | 7.0004 | 8.01965163019
$269$ | $\mathbb{B}_{269}^{22,11}$ |  1fffff800ff0699b6c1f21f0db3632786c6c69632731cb5a35ac7 1986b54aa955555 | 7.3092 | 7.4414849856
$270$ | $\mathbb{B}_{269}\circ\eta_{1}$ |  3fffff001fe0d336d83e43e1b66c64f0d8d8d2c64e6396b46b58e 330d6a9552aaaab | 7.0056 | 7.27399720615
$271$ | $\mathbb{B}_{271}^{21}$ |  7ffffe89502ef15b2327cc786c3c6784ad0fc5a64b4e5a4ca7373 812ef501deaaaaa | 7.5386 | 7.69015706806
$272$ | $\mathbb{B}_{271}\circ\eta_{1}$ |  fffffd12a05de2b6464f98f0d878cf095a1f8b4c969cb4994e6e7 025dea03bd55555 | - | 7.35719968178
$273$ | $\mathbb{B}_{273}^{22,11}$ |  1fffff800ff0338dbc2761b32787386c3e4e52c693696331a762d 1c932b54aa955555 | - | 7.21062306502
$274$ | $\mathbb{B}_{273}\circ\eta_{2}$ |  3fffff001fe0671b784ec3664f0e70d87c9ca58d26d2c6634ec5a 392656a9552aaaaa | - | 7.1843062201
$275$ | $\mathbb{B}_{275}^{22,11}$ |  7ffffe003f63918fc8c7a478f24e63c1e247694b66c72da47a4dc ad91b62b556aaaaa | - | 7.50099186669
$276$ | $\mathbb{B}_{275}\circ\eta_{1}$ |  fffffc007ec7231f918f48f1e49cc783c48ed296cd8e5b48f49b9 5b236c56aad55555 | - | 7.47703180212
$277$ | $\mathbb{B}_{275}\circ\eta_{1}\circ\eta_{5}$ |  1fffffc007ec7231f918f48f1e49cc783c48ed296cd8e5b48f49b 95b236c56aad55555 | - | 7.45520792849
$278$ | $\mathbb{B}_{275}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}$ |  3fffff800fd8e463f231e91e3c9398f07891da52d9b1cb691e937 2b646d8ad55aaaaaa | - | 7.12557624931

TABLE VI: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$279$ |  $\mathbb{B}_{275}\circ\eta_{1}\circ\eta_{2}\circ\eta_{1}\circ\eta_{5}$ |  7fffff001fb1c8c7e463d23c792731e0f123b4a5b36396d23d26e 56c8db15aab555555 | - | 7.05209277043
$280$ |  $\mathbb{B}_{281}\circ\eta_{4}$ |  fffffc007936c187ec0dd8f03db13cda71b39278cb138b52d88d4 ea594e31a554aaaaa | - | 7.27002967359
$281$ |  $\mathbb{B}_{281}^{22,11}$ |  1fffff800f26d830fd81bb1e07b6279b4e36724f1962716a5b11a 9d4b29c634aa955555 | 7.5058 | 7.7050156128
$282$ |  $\mathbb{B}_{281}\circ\eta_{2}$ |  3fffff001e4db061fb03763c0f6c4f369c6ce49e32c4e2d4b6235 3a96538c69552aaaaa | - | 7.38933283776
$283$ |  $\mathbb{B}_{283}^{22,11}$ |  7ffffe003f9c0f3c38d867139330e53e47a17a46b06d331b12658 db4b2d49ab556aaaaa | 7.5088 | 8.17067945317
$284$ |  $\mathbb{B}_{283}\circ\eta_{1}$ |  fffffc007f381e7871b0ce272661ca7c8f42f48d60da663624cb1 b6965a9356aad55555 | - | 7.89815902859
$285$ |  $\mathbb{B}_{285}^{22,11}$ |  1fffff800ff218fc31e07b06d24f26c6c6d999c6c6c634e3c6b16 a5b2d49a354aa955555 | 7.0142 | 7.46827877896
$286$ |  $\mathbb{B}_{285}\circ\eta_{1}$ |  3fffff001fe431f863c0f60da49e4d8d8db3338d8d8c69c78d62d 4b65a9346a9552aaaab | - | 7.22707192083
$287$ |  $\mathbb{B}_{287}^{22,11}$ |  7ffffe003fe06d8db331e63e16e4b4c78c6d0e4da4c3c6e16b669 3338d8e56ab556aaaaa | - | 7.47314461985
$288$ |  $\mathbb{B}_{287}\circ\eta_{1}$ |  fffffc007fc0db1b6663cc7c2dc9698f18da1c9b49878dc2d6cd2 6671b1cad56aad55555 | - | 7.21503131524
$289$ |  $\mathbb{B}_{289}^{22,11,6}$ |  1fffff800ff619ccc3c4e6723d3a5e35b0f24e34b1f25e13d2366 4ed2ccd9a754aa955555 | - | 7.1312329235
$290$ |  $\mathbb{B}_{291}\circ\eta_{4}$ |  3fffff001fec4a76992c5a7072d0cd969cc9b18cd879ccbc36b61 ec398760ec55aab55555 | - | 7.36040609137
$291$ |  $\mathbb{B}_{291}^{22,11,6}$ |  7ffffe003fd894ed3258b4e0e5a19b2d39936319b0f399786d6c3 d8730ec1d8ab556aaaaa | - | 7.71370012753
$292$ |  $\mathbb{B}_{291}\circ\eta_{1}$ |  fffffc007fb129da64b169c1cb43365a7326c63361e732f0dad87 b0e61d83b156aad55555 | - | 7.58846564614
$293$ |  $\mathbb{B}_{291}\circ\eta_{1}\circ\eta_{5}$ |  1fffffc007fb129da64b169c1cb43365a7326c63361e732f0dad8 7b0e61d83b156aad55555 | - | 7.47032718413
$294$ |  $\mathbb{B}_{291}\circ\eta_{1}\circ\eta_{1}\circ\eta_{5}$ |  3fffff800ff6253b4c962d38396866cb4e64d8c66c3ce65e1b5b0 f61cc3b0762ad55aaaaab | - | 7.33129770992
$295$ |  $\mathbb{B}_{295}^{22,11,6}$ |  7ffffe003fc319e87b0f1cc672969a589c69c49e49d879e1f264c 92d3a5e9934ab556aaaaa | - | 7.28730530899
$296$ |  $\mathbb{B}_{295}\circ\eta_{1}$ |  fffffc007f8633d0f61e398ce52d34b138d3893c93b0f3c3e4c99 25a74bd326956aad55555 | - | 7.13950456323
$297$ |  $\mathbb{B}_{297}^{22,11,6}$ |  1fffff800fce123d389c6a5e972c63932799399399633926c3785 e06d893d23a4d4aa955555 | - | 7.1551752109
$298$ |  $\mathbb{B}_{297}\circ\eta_{2}$ |  3fffff001f9c247a7138d4bd2e58c7264f32732732c6724d86f0b c0db127a4749a9552aaaaa | - | 7.26829268293
$299$ |  $\mathbb{B}_{297}\circ\eta_{2}\circ\eta_{5}$ |  7fffff001f9c247a7138d4bd2e58c7264f32732732c6724d86f0b c0db127a4749a9552aaaaa | - | 7.38485048736
$300$ |  $\mathbb{B}_{301}\circ\eta_{4}$ |  b7e00048d23dbb673e05a41e139b4cb590bd183cc39b16947856b 263b8b70dc5556a3d55 | - | 7.67132628708
$301$ |  $\mathbb{B}_{301}^{12}$ |  16fc00091a47b76ce7c0b483c2736996b217a307987362d28f0ad 64c7716e1b8aaad47aaa | 7.7173 | 8.24544958136
$302$ |  $\mathbb{B}_{301}\circ\eta_{1}$ |  2df80012348f6ed9cf81690784e6d32d642f460f30e6c5a51e15a c98ee2dc371555a8f555 | - | 8.092635315
$303$ |  $\mathbb{B}_{301}\circ\eta_{1}\circ\eta_{1}$ |  5bf00024691eddb39f02d20f09cda65ac85e8c1e61cd8b4a3c2b5 931dc5b86e2aab51eaab | 7.9488 | 8.16370264983
$304$ |  $\mathbb{B}_{301}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}$ |  b7e00048d23dbb673e05a41e139b4cb590bd183cc39b16947856b 263b8b70dc5556a3d557 | 7.0353 | 8.24553890079
$305$ |  $\mathbb{B}_{301}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}$ |  16fc00091a47b76ce7c0b483c2736996b217a307987362d28f0ad 64c7716e1b8aaad47aaae | 7.5117 | 8.12587351502
$306$ |  $\mathbb{B}_{301}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{6}\circ\eta_{5}$ |  2000b7e00048d23dbb673e05a41e139b4cb590bd183cc39b16947 856b263b8b70dc5556a3d557 | - | 8.18353434714
$307$ |  $\mathbb{B}_{301}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{6}\circ\eta_{5}\circ\eta_{6}$ |  2000b7e00048d23dbb673e05a41e139b4cb590bd183cc39b16947 856b263b8b70dc5556a3d557 | 7.4932 | 8.27180972442
$308$ |  $\mathbb{B}_{301}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}\circ\eta_{6}\circ\eta_{5}\circ\eta_{6}$ |  40016fc00091a47b76ce7c0b483c2736996b217a307987362d28f 0ad64c7716e1b8aaad47aaae | - | 8.15823873409
$309$ |  $\mathbb{B}_{301}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}\circ\eta_{2}\circ\eta_{6}\circ\eta_{5}\circ\eta_{6}$ |  8002df80012348f6ed9cf81690784e6d32d642f460f30e6c5a51e 15ac98ee2dc371555a8f555c | 7.5229 | 8.08338977311
$310$ |  $\mathbb{B}_{309}\circ\eta_{6}$ |  8002df80012348f6ed9cf81690784e6d32d642f460f30e6c5a51e 15ac98ee2dc371555a8f555c | - | 7.96716962361
$311$ |  $\mathbb{B}_{309}\circ\eta_{6}\circ\eta_{5}$ |  48002df80012348f6ed9cf81690784e6d32d642f460f30e6c5a51 e15ac98ee2dc371555a8f555c | 7.4229 | 7.88786494862
$312$ |  $\mathbb{B}_{309}\circ\eta_{2}\circ\eta_{6}\circ\eta_{5}$ |  90005bf00024691eddb39f02d20f09cda65ac85e8c1e61cd8b4a3 c2b5931dc5b86e2aab51eaab8 | - | 7.69152970923

TABLE VII: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$313$ |  $\mathbb{B}_{313}^{24,11,9,4}$ |  1fffffe003fe04b878f0b32666cda7c2c5a4f98f4994e1ec2d61c c666330b49690ea552aa555555 | 7.5547 | 7.62048848787
$314$ |  $\mathbb{B}_{313}\circ\eta_{1}$ |  3fffffc007fc0970f1e1664ccd9b4f858b49f31e9329c3d85ac39 8ccc661692d21d4aa554aaaaab | - | 7.31315828512
$315$ |  $\mathbb{B}_{315}^{23,10,7,4}$ |  7fffff003f87c0f184fe339cf0c6179e38de666466668db69a164 d2c9b36ac592d4a5ab552aaaaa | 7.4661 | 7.5204638472
$316$ |  $\mathbb{B}_{315}\circ\eta_{2}$ |  fffffe007f0f81e309fc6739e18c2f3c71bcccc8cccd1b6d342c9 a59366d58b25a94b56aa555554 | - | 7.25908694388
$317$ |  $\mathbb{B}_{317}$ |  1ffffff003f920d9f36093b4ec686396db0f225e25e234b1c7926 86c4f138a7359ca3952ab555555 | - | 7.46131571132
$318$ |  $\mathbb{B}_{317}\circ\eta_{2}$ |  3fffffe007f241b3e6c12769d8d0c72db61e44bc4bc469638f24d 0d89e2714e6b39472a556aaaaaa | - | 7.23658222413
$319$ |  $\mathbb{B}_{319}^{25,10,7}$ |  7fffffc00fe0674e1939ce09ccf92963cd83c378da34b58cb61f1 acc9d6c9b196c2656ad54aaaaaa | 7.4224 | 7.45720357614
$320$ |  $\mathbb{B}_{319}\circ\eta_{2}$ |  ffffff801fc0ce9c32739c1399f252c79b0786f1b4696b196c3e3 5993ad93632d84cad5aa9555554 | - | 7.48976009362
$321$ |  $\mathbb{B}_{323}\circ\eta_{0}$ |  1fffffc003f9807e330e1ce9c63d0e61e0cd83c9998d29cca5a64 bd26d84da4b3256a9952aad55555 | 7.3183 | 7.73116746699
$322$ |  $\mathbb{B}_{323}\circ\eta_{4}$ |  3fffffc003f9807e330e1ce9c63d0e61e0cd83c9998d29cca5a64 bd26d84da4b3256a9952aad55555 | - | 7.76891952645
$323$ |  $\mathbb{B}_{323}^{24,12,7}$ |  7fffff8007f300fc661c39d38c7a1cc3c19b0793331a53994b4c9 7a4db09b49664ad532a555aaaaaa | 7.743 | 7.80788804071
$324$ |  $\mathbb{B}_{323}\circ\eta_{2}$ |  ffffff000fe601f8cc3873a718f4398783360f266634a73296992 f49b613692cc95aa654aab555554 | - | 7.68491947291
$325$ |  $\mathbb{B}_{323}\circ\eta_{2}\circ\eta_{1}$ |  1fffffe001fcc03f19870e74e31e8730f066c1e4ccc694e652d32 5e936c26d25992b54ca9556aaaaa9 | 7.5167 | 7.61206399539
$326$ |  $\mathbb{B}_{323}\circ\eta_{2}\circ\eta_{1}\circ\eta_{6}$ |  1fffffe001fcc03f19870e74e31e8730f066c1e4ccc694e652d32 5e936c26d25992b54ca9556aaaaa9 | - | 7.50642746151
$327$ |  $\mathbb{B}_{327}^{25,12,11,6}$ |  7fffffc003ff81ec63781ce43c19c39e1e66786619665a666969b 4994b46c95a364e95aab554aaaaaa | 7.3009 | 7.76761586518
$328$ |  $\mathbb{B}_{323}\circ\eta_{2}\circ\eta_{1}\circ\eta_{1}\circ\eta_{6}\circ\eta_{6}$ |  3fffffc003f9807e330e1ce9c63d0e61e0cd83c9998d29cca5a64 bd26d84da4b3256a9952aad555553 | - | 7.22622246104
$329$ |  $\mathbb{B}_{327}\circ\eta_{1}\circ\eta_{2}$ |  1ffffff000ffe07b18de07390f0670e787999e19865996999a5a6 d2652d1b2568d93a56aad552aaaaaa | 7.2782 | 7.33340108401
$330$ |  $\mathbb{B}_{327}\circ\eta_{1}\circ\eta_{2}\circ\eta_{1}$ |  3fffffe001ffc0f631bc0e721e0ce1cf0f333c330cb32d3334b4d a4ca5a364ad1b274ad55aaa5555555 | - | 7.22819593787
$331$ |  $\mathbb{B}_{331}^{24,12,8,2}$ |  7fffff8007f83c0fcc36f0de9667261b24cb1a5b63638793cc739 672661e8d2e34cad4b5aa555aaaaaa | 7.3501 | 7.44603778714
$332$ |  $\mathbb{B}_{331}\circ\eta_{2}$ |  ffffff000ff0781f986de1bd2cce4c36499634b6c6c70f2798e72 ce4cc3d1a5c6995a96b54aab555554 | - | 7.2117246794
$333$ |  $\mathbb{B}_{333}^{24,12,8,3}$ |  1fffffe001fe3181fc303c963e42f18cc6c63a670b70b66126c6c c9b42e5278d2b2d5a9b255aaa555555 | 7.2743 | 7.29724927613
$334$ |  $\mathbb{B}_{333}\circ\eta_{2}$ |  3fffffc003fc6303f860792c7c85e3198d8c74ce16e16cc24d8d9 93685ca4f1a565ab5364ab554aaaaaa | - | 7.06855911798
$335$ |  $\mathbb{B}_{335}^{25,12,8,2}$ |  7fffffc003fc81e67b3698760f1b0f0cb46e46b61e6963e46e43c d2d392d6259e33a6695cab554aaaaaa | 7.3302 | 7.46276100545
$336$ |  $\mathbb{B}_{335}\circ\eta_{2}$ |  ffffff8007f903ccf66d30ec1e361e1968dc8d6c3cd2c7c8dc879 a5a725ac4b3c674cd2b956aa9555554 | - | 7.27422680412
$337$ |  $\mathbb{B}_{337}^{25,12,8,2}$ |  1ffffff000ff219cc6f03790f81e4f25a6931cb399b19930db386 1e34e5a94b972b46cd9a354aab555555 | 7.3327 | 7.35550518135
$338$ |  $\mathbb{B}_{337}\circ\eta_{2}$ |  3fffffe001fe43398de06f21f03c9e4b4d26396733633261b670c 3c69cb52972e568d9b346a9556aaaaaa | - | 7.26280991736
$339$ |  $\mathbb{B}_{339}^{26,13,10,2}$ |  7fffffe000ffc231ec37835a670f85a749ccd8db61b9638d8cc9c 2785ad267835a34e9374aad556aaaaaa | 7.3961 | 7.77228459353
$340$ |  $\mathbb{B}_{339}\circ\eta_{1}$ |  ffffffc001ff8463d86f06b4ce1f0b4e9399b1b6c372c71b19938 4f0b5a4cf06b469d26e955aaad555555 | - | 7.7500670421
$341$ |  $\mathbb{B}_{339}\circ\eta_{1}\circ\eta_{5}$ |  1ffffffc001ff8463d86f06b4ce1f0b4e9399b1b6c372c71b1993 84f0b5a4cf06b469d26e955aaad555555 | 7.317 | 7.72939377825
$342$ |  $\mathbb{B}_{339}\circ\eta_{1}\circ\eta_{1}\circ\eta_{5}$ |  3ffffff8003ff08c7b0de0d699c3e169d2733636d86e58e363327 09e16b499e0d68d3a4dd2ab555aaaaaab | - | 7.67984241628
$343$ |  $\mathbb{B}_{339}\circ\eta_{1}\circ\eta_{1}\circ\eta_{5}\circ\eta_{6}$ |  3ffffff8003ff08c7b0de0d699c3e169d2733636d86e58e363327 09e16b499e0d68d3a4dd2ab555aaaaaab | 7.1921 | 7.63260672116
$344$ |  $\mathbb{B}_{339}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{5}\circ\eta_{6}$ |  7ffffff0007fe118f61bc1ad3387c2d3a4e66c6db0dcb1c6c664e 13c2d6933c1ad1a749ba556aab5555557 | - | 7.36286709806
$345$ |  $\mathbb{B}_{345}^{27,13,10,6}$ |  1ffffffc001ff8187e1f2364edc38f18e730f933a53a13e13394b 3649b492dc4e7235a569a955aaad555555 | 7.2612 | 7.48773276296

TABLE VIII: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$346$ |  $\mathbb{B}_{339}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}\circ\eta_{6}$ |  1ffffffc001ff8463d86f06b4ce1f0b4e9399b1b6c372c71b1993 84f0b5a4cf06b469d26e955aaad555555e | - | 7.10818192614
$347$ |  $\mathbb{B}_{347}^{26,13,10,0}$ |  7fffffe000ffcc0f1e19c327c33966f03f21d8e17a6367a16d897 2b52e61b34a73499692d4caad556aaaaaa | 7.1698 | 7.23177177177
$348$ |  $\mathbb{B}_{347}\circ\eta_{2}$ |  ffffffc001ff981e3c33864f8672cde07e43b1c2f4c6cf42db12e 56a5cc36694e6932d25a9955aaad555554 | - | 7.13888233907
$349$ |  $\mathbb{B}_{349}^{24,9,9,9}$ |  1fffffe00ff8033e76c49b0a79274c36a5ad0f0399b3931992b4b c1e072cf63960b18ec76532a954aa555555 | 7.1295 | 7.24488460623
$350$ |  $\mathbb{B}_{349}\circ\eta_{2}$ |  3fffffc01ff0067ced893614f24e986d4b5a1e073367263325697 83c0e59ec72c1631d8eca6552a954aaaaaa | - | 7.14619064287
$351$ |  $\mathbb{B}_{349}\circ\eta_{2}\circ\eta_{1}$ |  7fffff803fe00cf9db126c29e49d30da96b43c0e66ce4c664ad2f 0781cb3d8e582c63b1d94caa552a9555555 | 7.0911 | 7.14043120436
$352$ |  $\mathbb{B}_{349}\circ\eta_{2}\circ\eta_{1}\circ\eta_{5}$ |  ffffff803fe00cf9db126c29e49d30da96b43c0e66ce4c664ad2f 0781cb3d8e582c63b1d94caa552a9555555 | - | 7.05603644647
$353$ |  $\mathbb{B}_{353}^{23,9,9,9}$ |  1fffffc01ff006cf9f21e06d8e74c72667963c1b8b6b4f07091ad 27966636cf649c6a5a3594c6ab55aad55555 | 7.1385 | 7.26498367537
$354$ |  $\mathbb{B}_{355}\circ\eta_{4}$ |  3fffff803fe00f39e1b8784d2f26ccd9cc63c0f307c24e2d6b34a d26cd9ccc6343ce9691a5934aa552a955555 | - | 7.18472652219
$355$ |  $\mathbb{B}_{355}^{23,9,9,9}$ |  7fffff007fc01e73c370f09a5e4d99b398c781e60f849c5ad6695 a4d9b3998c6879d2d234b26954aa552aaaaa | 7.232 | 7.39496537965
$356$ |  $\mathbb{B}_{355}\circ\eta_{2}$ |  fffffe00ff803ce786e1e134bc9b3367318f03cc1f0938b5acd2b 49b3673318d0f3a5a46964d2a954aa555554 | - | 7.20582215147
$357$ |  $\mathbb{B}_{357}^{26,12,6}$ |  1ffffff8007f876cc33e58e06f198373723c78d293d29983d383c 96d23737299b46a49e532cc76956aa9555555 | 7.2201 | 7.48467230444
$358$ |  $\mathbb{B}_{357}\circ\eta_{2}$ |  3ffffff000ff0ed9867cb1c0de3306e6e478f1a527a53307a7079 2da46e6e53368d493ca6598ed2ad552aaaaaa | - | 7.27625752243
$359$ |  $\mathbb{B}_{357}\circ\eta_{2}\circ\eta_{2}$ |  7fffffe001fe1db30cf96381bc660dcdc8f1e34a4f4a660f4e0f2 5b48dcdca66d1a92794cb31da55aaa5555554 | 7.1896 | 7.33361784454
$360$ |  $\mathbb{B}_{357}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}$ |  ffffffe001fe1db30cf96381bc660dcdc8f1e34a4f4a660f4e0f2 5b48dcdca66d1a92794cb31da55aaa5555554 | - | 7.12714474263
$361$ |  $\mathbb{B}_{357}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}\circ\eta_{6}$ |  ffffffe001fe1db30cf96381bc660dcdc8f1e34a4f4a660f4e0f2 5b48dcdca66d1a92794cb31da55aaa5555554 | 7.1229 | 7.17310656099
$362$ |  $\mathbb{B}_{363}\circ\eta_{4}$ |  3ffffff000fde381fac783318cb427c3396999ce1ccbc4ed0cda4 d9987932d62f0c9b3296c15a925d4aab555555 | - | 7.86295451818
$363$ |  $\mathbb{B}_{363}^{26,12,6}$ |  7fffffe001fbc703f58f066319684f8672d3339c399789da19b49 b330f265ac5e1936652d82b524ba9556aaaaaa | 7.6 | 7.92929353713
$364$ |  $\mathbb{B}_{363}\circ\eta_{1}$ |  ffffffc003f78e07eb1e0cc632d09f0ce5a66738732f13b433693 6661e4cb58bc326cca5b056a49752aad555555 | - | 7.68003709715
$365$ |  $\mathbb{B}_{363}\circ\eta_{1}\circ\eta_{2}$ |  1ffffff8007ef1c0fd63c198c65a13e19cb4cce70e65e276866d2 6ccc3c996b17864d994b60ad492ea555aaaaaaa | 7.2421 | 7.46274927179
$366$ |  $\mathbb{B}_{363}\circ\eta_{1}\circ\eta_{2}\circ\eta_{2}$ |  3ffffff000fde381fac783318cb427c3396999ce1ccbc4ed0cda4 d9987932d62f0c9b3296c15a925d4aab5555554 | - | 7.43291532571
$367$ |  $\mathbb{B}_{363}\circ\eta_{1}\circ\eta_{2}\circ\eta_{2}\circ\eta_{2}$ |  7fffffe001fbc703f58f066319684f8672d3339c399789da19b49 b330f265ac5e1936652d82b524ba9556aaaaaa8 | 7.0216 | 7.28600021638
$368$ |  $\mathbb{B}_{363}\circ\eta_{1}\circ\eta_{2}\circ\eta_{2}\circ\eta_{2}\circ\eta_{2}$ |  ffffffc003f78e07eb1e0cc632d09f0ce5a66738732f13b433693 6661e4cb58bc326cca5b056a49752aad5555550 | - | 7.07692307692
$369$ |  $\mathbb{B}_{363}\circ\eta_{5}\circ\eta_{5}\circ\eta_{6}\circ\eta_{5}\circ\eta_{6}\circ\eta_{6}$ |  5ffffffe001fbc703f58f066319684f8672d3339c399789da19b4 9b330f265ac5e1936652d82b524ba9556aaaaaa | 7.074 | 7.19667019027
$370$ |  $\mathbb{B}_{369}\circ\eta_{6}$ |  5ffffffe001fbc703f58f066319684f8672d3339c399789da19b4 9b330f265ac5e1936652d82b524ba9556aaaaaa | - | 7.02988600185
$371$ |  $\mathbb{B}_{371}^{23,9,9,9}$ |  7fffff007fc01f9de489b87b439839334cc34a5a736938cdb1e32 787c34cc331b59b43a5b9dc689a954aa552aaaaa | 7.333 | 7.44730007575
$372$ |  $\mathbb{B}_{371}\circ\eta_{1}$ |  fffffe00ff803f3bc91370f687307266998694b4e6d2719b63c64 f0f869986636b36874b73b8d1352a954aa555555 | - | 7.53561315618
$373$ |  $\mathbb{B}_{371}\circ\eta_{1}\circ\eta_{5}$ |  1fffffe00ff803f3bc91370f687307266998694b4e6d2719b63c6 4f0f869986636b36874b73b8d1352a954aa555555 | 7.2147 | 7.62601403201
$374$ |  $\mathbb{B}_{371}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}$ |  3fffffc01ff007e779226e1ed0e60e4cd330d2969cda4e336c78c 9e1f0d330cc6d66d0e96e771a26a552a954aaaaaa | - | 7.51133068414
$375$ |  $\mathbb{B}_{371}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}\circ\eta_{5}$ |  7fffffc01ff007e779226e1ed0e60e4cd330d2969cda4e336c78c 9e1f0d330cc6d66d0e96e771a26a552a954aaaaaa | 7.0011 | 7.40209495736
$376$ |  $\mathbb{B}_{371}\circ\eta_{1}\circ\eta_{2}\circ\eta_{2}\circ\eta_{1}\circ\eta_{5}$ |  ffffff007fc01f9de489b87b439839334cc34a5a736938cdb1e32 787c34cc331b59b43a5b9dc689a954aa552aaaaa9 | - | 7.28141738772
$377$ |  $\mathbb{B}_{377}^{27,13,10,5}$ |  1ffffffc001ff80e783ce03cce4f872786f07b0db0d99c6666d99 cb1cb16b46963694e4cd2a4d2964a955aaad555555 | 7.2249 | 7.35048613984
$378$ |  $\mathbb{B}_{377}\circ\eta_{2}$ |  3ffffff8003ff01cf079c0799c9f0e4f0de0f61b61b338cccdb33 963962d68d2c6d29c99a549a52c952ab555aaaaaaa | - | 7.11786390356
$379$ |  $\mathbb{B}_{379}^{27,13,10,5}$ |  7ffffff0007fe0761e4f90f21fc30f06d8d86cf09b319a6367993 39d2ce58d8e52d34a972d1ac696256aa5552aaaaaa | 7.2422 | 7.33685769742
$380$ |  $\mathbb{B}_{379}\circ\eta_{1}\circ\eta_{1}\circ\eta_{3}$ |  ffffffc001ff81d8793e43c87f0c3c1b6361b3c26cc6698d9e64c e74b39636394b4d2a5cb46b1a5895aa9554aaaaaab | - | 7.01924946529

TABLE IX: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$381$ |  $\mathbb{B}_{381}^{27,13,10,5}$ |  1ffffffc001ff818f0787c0d9e13e0cf0e49e43cc39c667333666 d92cd2e58e4b4ca53a59cad696b49a955aaad555555 | 7.106 | 7.2018753721
$382$ |  $\mathbb{B}_{381}\circ\eta_{2}$ |  3ffffff8003ff031e0f0f81b3c27c19e1c93c8798738cce666ccd b259a5cb1c96994a74b395ad2d69352ab555aaaaaaa | - | 7.08437712399
$383$ |  $\mathbb{B}_{383}^{27,13,10,5}$ |  7ffffff0007fe07319cce478e730f318f01bc1e3963c3f27272b4 b61b694b952d932d326da46cc993256aa5552aaaaaa | 7.0314 | 7.07480466866
$384$ |  $\mathbb{B}_{383}\circ\eta_{1}\circ\eta_{2}\circ\eta_{3}$ |  ffffffc001ff81cc673391e39cc3cc63c06f078e58f0fc9c9cad2 d86da52e54b64cb4c9b691b3264c95aa9554aaaaaaa | - | 7.08923076923
$385$ |  $\mathbb{B}_{385}^{27,13,10,7}$ |  1ffffffc001ff80d8c72d86c73343b439960f721f8d2d61b331a7 c3c95a374a7992f12f336c69c36c9ca955aaad555555 | 7.0772 | 7.24887519562
$386$ |  $\mathbb{B}_{385}\circ\eta_{2}$ |  3ffffff8003ff01b18e5b0d8e668768732c1ee43f1a5ac366634f 8792b46e94f325e25e66d8d386d93952ab555aaaaaaa | - | 7.1296774811
$387$ |  $\mathbb{B}_{387}^{27,13,10,5}$ |  7ffffff0007fe033c1f66139c2d327c96786f078d8dc39b33339b 48d8da52e5a61ca730f49b166294b356aa5552aaaaaa | 7.1502 | 7.19974040958
$388$ |  $\mathbb{B}_{387}\circ\eta_{1}$ |  ffffffe000ffc06783ecc27385a64f92cf0de0f1b1b8736666736 91b1b4a5cb4c394e61e9362cc52966ad54aaa5555555 | - | 7.06248827172
$389$ |  $\mathbb{B}_{391}\circ\eta_{0}$ |  1ffffff8003ff01c39c3e47876c1da64f931b70cda1e723633272 365a1ccb71b394e61dac7696e52d92dab552aa9555555 | 7.0461 | 7.39305256987
$390$ |  $\mathbb{B}_{391}\circ\eta_{4}$ |  3ffffff8003ff01c39c3e47876c1da64f931b70cda1e723633272 365a1ccb71b394e61dac7696e52d92dab552aa9555555 | - | 7.27403156385
$391$ |  $\mathbb{B}_{391}^{27,13,10,7,1}$ |  7ffffff0007fe0387387c8f0ed83b4c9f2636e19b43ce46c664e4 6cb43996e36729cc3b58ed2dca5b25b56aa5552aaaaaa | 7.1553 | 7.16070257611
$392$ |  $\mathbb{B}_{391}\circ\eta_{1}$ |  ffffffe000ffc070e70f91e1db076993e4c6dc336879c8d8cc9c8 d968732dc6ce539876b1da5b94b64b6ad54aaa5555555 | - | 7.1832460733
$393$ |  $\mathbb{B}_{391}\circ\eta_{1}\circ\eta_{2}$ |  1ffffffc001ff80e1ce1f23c3b60ed327c98db866d0f391b19939 1b2d0e65b8d9ca730ed63b4b7296c96d5aa9554aaaaaaa | 7.0952 | 7.19304210134
$394$ |  $\mathbb{B}_{391}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}$ |  3ffffffc001ff80e1ce1f23c3b60ed327c98db866d0f391b19939 1b2d0e65b8d9ca730ed63b4b7296c96d5aa9554aaaaaaa | - | 7.22095078612
$395$ |  $\mathbb{B}_{391}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}\circ\eta_{5}$ |  7ffffffc001ff80e1ce1f23c3b60ed327c98db866d0f391b19939 1b2d0e65b8d9ca730ed63b4b7296c96d5aa9554aaaaaaa | 7.0991 | 7.23610982284
$396$ |  $\mathbb{B}_{395}\circ\eta_{1}$ |  ffffffe000ffc0664e333c1e1e703721f9927c19cf1a65b0f34b1 e61b4d9ad63995a372b65a5ad3324e66ad54aaa5555555 | - | 7.15270935961
$397$ |  $\mathbb{B}_{397}^{27,13,8}$ |  1ffffffc001ff9213e36cf18664c8978353c49d2c9da358e948f8 49f21d8c3d8ed3f29788ce669b4c7253a3955aaad555555 | 7.0829 | 7.19675799087
$398$ |  $\mathbb{B}_{399}\circ\eta_{4}$ |  3ffffff8003fe69c336f097b072cd296cf04ed8b53ad278d99999 c963c13f09c4eb4c783cc36b178b4732d86552aa9555555 | - | 7.12056099973
$399$ |  $\mathbb{B}_{399}^{27,13,8}$ |  7ffffff0007fcd3866de12f60e59a52d9e09db16a75a4f1b33333 92c7827e1389d698f07986d62f168e65b0caa5552aaaaaa | 7.1487 | 7.23180703189
$400$ |  $\mathbb{B}_{399}\circ\eta_{1}$ |  ffffffe000ff9a70cdbc25ec1cb34a5b3c13b62d4eb49e3666667 258f04fc2713ad31e0f30dac5e2d1ccb61954aaa5555555 | - | 7.10479573712
$401$ |  $\mathbb{B}_{401}^{27,13,9}$ |  1ffffffc001ff901e1c3cd32370b1b0f16e06e4392664db258b09 e31ce66392e46a47b4b1b0b7233cd2da5ab955aaad555555 | 7.0084 | 7.01820006983
$402$ |  $\mathbb{B}_{407}\circ\eta_{0}\circ\eta_{0}\circ\eta_{4}$ |  3fffffe001fc8787c06f0f067337903d892d23cc1a79db198e764 99b1d961acd23c389d2b973366b4b46ad6968d5aaa555555 | - | 7.05755961219
$403$ |  $\mathbb{B}_{407}\circ\eta_{0}\circ\eta_{0}$ |  7fffffc003f90f0f80de1e0ce66f207b125a479834f3b6331cec9 3363b2c359a478713a572e66cd6968d5ad2d1ab554aaaaaa | - | 7.10264147643
$404$ |  $\mathbb{B}_{407}\circ\eta_{0}\circ\eta_{4}$ |  ffffffc003f90f0f80de1e0ce66f207b125a479834f3b6331cec9 3363b2c359a478713a572e66cd6968d5ad2d1ab554aaaaaa | - | 7.02911283376
$405$ |  $\mathbb{B}_{407}\circ\eta_{4}\circ\eta_{4}$ |  1ffffffc003f90f0f80de1e0ce66f207b125a479834f3b6331cec 93363b2c359a478713a572e66cd6968d5ad2d1ab554aaaaaa | - | 7.09082656061
$406$ |  $\mathbb{B}_{407}\circ\eta_{4}$ |  3ffffff8007f21e1f01bc3c19ccde40f624b48f3069e76c6639d9 266c76586b348f0e274ae5ccd9ad2d1ab5a5a356aa9555555 | - | 7.03765690377
$407$ |  $\mathbb{B}_{407}^{27,12,6}$ |  7ffffff000fe43c3e0378783399bc81ec49691e60d3ced8cc73b2 4cd8ecb0d6691e1c4e95cb99b35a5a356b4b46ad552aaaaaa | - | 7.11856467555
$408$ |  $\mathbb{B}_{407}\circ\eta_{2}$ |  ffffffe001fc8787c06f0f067337903d892d23cc1a79db198e764 99b1d961acd23c389d2b973366b4b46ad6968d5aaa5555554 | - | 7.1801242236
$409$ |  $\mathbb{B}_{407}\circ\eta_{2}\circ\eta_{6}$ |  ffffffe001fc8787c06f0f067337903d892d23cc1a79db198e764 99b1d961acd23c389d2b973366b4b46ad6968d5aaa5555554 | - | 7.24285590578
$410$ |  $\mathbb{B}_{407}\circ\eta_{2}\circ\eta_{2}\circ\eta_{6}$ |  1ffffffc003f90f0f80de1e0ce66f207b125a479834f3b6331cec 93363b2c359a478713a572e66cd6968d5ad2d1ab554aaaaaa8 | - | 7.0767028711
$411$ |  $\mathbb{B}_{411}^{28,14,11,8,4}$ |  7ffffff8001ffc03c6b678721f0d30dce4da41f861d984f2399c9 9b72c598965a9478c6c8d30d29725a63e4b54aa9555aaaaaaa | - | 7.13047699451
$412$ |  $\mathbb{B}_{411}\circ\eta_{1}$ |  fffffff0003ff8078d6cf0e43e1a61b9c9b483f0c3b309e473393 36e58b312cb528f18d91a61a52e4b4c7c96a9552aab5555555 | - | 7.01537444206
$413$ |  $\mathbb{B}_{413}^{28,14,10,6}$ |  1ffffffe0007fe06f0cfc3189e43ce49c9c9c333c0fc3c338cf0c b4c932d2d4ad332d8d8d8e4d2e589b2d4cb46a556aaa5555555 | - | 7.02855612329
$414$ |  $\mathbb{B}_{415}\circ\eta_{4}$ |  3ffffffc000ffc0783ccf807b61bc3c9e4b198d398721b4a76699 86760f1a36993c99b0e58d2d1a716a94cd296ad54aaad555555 | - | 7.09714285714

TABLE X: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$415$ |  $\mathbb{B}_{415}^{28,14,10,6}$ |  7ffffff8001ff80f0799f00f6c378793c96331a730e43694ecd33 0cec1e346d32793361cb1a5a34e2d5299a52d5aa9555aaaaaaa | - | 7.25891427126
$416$ |  $\mathbb{B}_{415}\circ\eta_{1}$ |  fffffff0003ff01e0f33e01ed86f0f2792c6634e61c86d29d9a66 19d83c68da64f266c39634b469c5aa5334a5ab552aab5555555 | - | 7.04855001629
$417$ |  $\mathbb{B}_{417}^{28,13,12,8,6,6,4}$ |  1ffffffe000fff00fc0fc92787993c3e0793934f867264730c739 936cb36e636694f39396a52d39969638d4ad4ab554aaa5555555 | - | 7.22610538564
$418$ |  $\mathbb{B}_{417}\circ\eta_{2}$ |  3ffffffc001ffe01f81f924f0f32787c0f27269f0ce4c8e618e73 26d966dcc6cd29e7272d4a5a732d2c71a95a956aa9554aaaaaaa | - | 7.13683522588
$419$ |  $\mathbb{B}_{417}\circ\eta_{2}\circ\eta_{5}$ |  7ffffffc001ffe01f81f924f0f32787c0f27269f0ce4c8e618e73 26d966dcc6cd29e7272d4a5a732d2c71a95a956aa9554aaaaaaa | - | 7.05120893244
$420$ |  $\mathbb{B}_{421}\circ\eta_{4}$ |  fffffff0007ff807e07f031b2d86c9cc670cd8786d23c9c399b0c d399b49cb70e5a58cd264c9ce58f39352a56a55aaa5552aaaaaa | - | 7.04585397028
$421$ |  $\mathbb{B}_{421}^{28,13,12,8,6,6,4}$ |  1ffffffe000fff00fc0fe06365b0d9398ce19b0f0da4793873361 9a73369396e1cb4b19a4c9939cb1e726a54ad4ab554aaa5555555 | - | 7.25327385824
$422$ |  $\mathbb{B}_{421}\circ\eta_{1}$ |  3ffffffc001ffe01f81fc0c6cb61b27319c3361e1b48f270e66c3 34e66d272dc39696334993273963ce4d4a95a956aa9554aaaaaab | - | 7.05842251288
$423$ |  $\mathbb{B}_{423}^{28,14,11,8,4}$ |  7ffffff8001ffc03cc366f03d907b07993e4e1f21e1c3c66d2727 27270e64b49697296c6b19a53a518b52e634cb54aa9555aaaaaaa | - | 7.17150300601
$424$ |  $\mathbb{B}_{423}\circ\eta_{1}$ |  fffffff0003ff807986cde07b20f60f327c9c3e43c3878cda4e4e 4e4e1cc9692d2e52d8d6334a74a316a5cc6996a9552aab5555555 | - | 7.08448928121
$425$ |  $\mathbb{B}_{425}^{28,14,10,6}$ |  1ffffffe0007fe03c25bc93927f0bc3326799334f07e4c9d90db7 0b71cb9d8ce56b4f339966332d0b563938d1e2d2a556aaa555555 5 | - | 7.0667057903
$426$ |  $\mathbb{B}_{429}\circ\eta_{0}\circ\eta_{4}$ |  3ffffff8001ffc03f07e1330b4e0e721a5e24d8e664e34e1e49b0 dcb18e5a4f24e6649ce25e1a364a4f0b33a56b52ad55aaa955555 5 | - | 7.05528341498
$427$ |  $\mathbb{B}_{429}\circ\eta_{0}$ |  7ffffff0003ff807e0fc266169c1ce434bc49b1ccc9c69c3c9361 b9631cb49e49ccc939c4bc346c949e16674ad6a55aab5552aaaaa a | - | 7.22782050266
$428$ |  $\mathbb{B}_{429}\circ\eta_{4}$ |  fffffff0003ff807e0fc266169c1ce434bc49b1ccc9c69c3c9361 b9631cb49e49ccc939c4bc346c949e16674ad6a55aab5552aaaaa a | - | 7.13889321902
$429$ |  $\mathbb{B}_{429}^{28,14,11,8,4}$ |  1ffffffe0007ff00fc1f84cc2d3839c86978936399938d387926c 372c639693c93999273897868d9293c2cce95ad4ab556aaa55555 55 | - | 7.05354131535
$430$ |  $\mathbb{B}_{431}\circ\eta_{4}$ |  3ffffffc000ffe01e786f1c27b0cf61f0e633c1e9c3998ce46e52 d9c3e46e4c9992d85ad3264b5a74cb162db46965aa554aaad5555 55 | - | 7.0021964705
$431$ |  $\mathbb{B}_{431}^{28,14,11,8,4}$ |  7ffffff8001ffc03cf0de384f619ec3e1cc6783d3873319c8dca5 b387c8dc993325b0b5a64c96b4e9962c5b68d2cb54aa9555aaaaa aa | - | 7.08849118522
$432$ |  $\mathbb{B}_{431}\circ\eta_{1}$ |  fffffff0003ff8079e1bc709ec33d87c398cf07a70e663391b94b 670f91b932664b616b4c992d69d32c58b6d1a596a9552aab55555 55 | - | 7.04560555723
$433$ |  $\mathbb{B}_{433}^{28,13,12,8,6,6,4}$ |  1ffffffe000fff00fc0fc333247c1e1e0c7387999a5b8d3c619ce 1cda4d9a6d3c91e19996936ca5a5ad6e3332d4ad4ab554aaa5555 555 | - | 7.05482390126
$434$ |  $\mathbb{B}_{433}\circ\eta_{2}$ |  3ffffffc001ffe01f81f866648f83c3c18e70f3334b71a78c339c 39b49b34da7923c3332d26d94b4b5adc6665a95a956aa9554aaaa aaa | - | 7.06140811277
$435$ |  $\mathbb{B}_{433}\circ\eta_{2}\circ\eta_{1}$ |  7ffffff8003ffc03f03f0ccc91f0787831ce1e66696e34f186738 73693669b4f24786665a4db29696b5b8cccb52b52ad552aa95555 555 | - | 7.06854688084
$436$ |  $\mathbb{B}_{433}\circ\eta_{2}\circ\eta_{1}\circ\eta_{5}$ |  fffffff8003ffc03f03f0ccc91f0787831ce1e66696e34f186738 73693669b4f24786665a4db29696b5b8cccb52b52ad552aa95555 555 | - | 7.07307635065
$437$ |  $\mathbb{B}_{433}\circ\eta_{2}\circ\eta_{1}\circ\eta_{5}\circ\eta_{5}$ |  1fffffff8003ffc03f03f0ccc91f0787831ce1e66696e34f18673 873693669b4f24786665a4db29696b5b8cccb52b52ad552aa9555 5555 | - | 7.07816901408
$438$ |  $\mathbb{B}_{441}\circ\eta_{4}\circ\eta_{4}\circ\eta_{4}$ |  3ffffff0007fe0479c679c39c3873b1ce523c4e1f09784e46c3f0 cdc8cd2b4e46c5a1d296c4b706c93b25b49b49a649a456aa5552a aaaa | - | 7.01645819618
$439$ |  $\mathbb{B}_{441}\circ\eta_{0}$ |  7fffffc001ff811e719e70e70e1cec73948f1387c25e1391b0fc3 372334ad391b16874a5b12dc1b24ec96d26d269926915aa9554aa aaaa | - | 7.07544606799
$440$ |  $\mathbb{B}_{441}\circ\eta_{4}$ |  ffffffc001ff811e719e70e70e1cec73948f1387c25e1391b0fc3 372334ad391b16874a5b12dc1b24ec96d26d269926915aa9554aa aaaa | - | 7.16400236827
$441$ |  $\mathbb{B}_{441}^{26,13,10,6}$ |  1ffffff8003ff023ce33ce1ce1c39d8e7291e270f84bc272361f8 66e46695a72362d0e94b625b83649d92da4da4d324d22b552aa95 55555 | - | 7.25458818263

TABLE XI: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$442$ |  $\mathbb{B}_{441}\circ\eta_{2}$ |  3ffffff0007fe0479c679c39c3873b1ce523c4e1f09784e46c3f0 cdc8cd2b4e46c5a1d296c4b706c93b25b49b49a649a456aa5552a aaaaa | - | 7.07994491556
$443$ |  $\mathbb{B}_{441}\circ\eta_{2}\circ\eta_{1}$ |  7fffffe000ffc08f38cf3873870e7639ca4789c3e12f09c8d87e1 9b919a569c8d8b43a52d896e0d92764b6936934c9348ad54aaa55 55555 | - | 7.06694274397
$444$ |  $\mathbb{B}_{441}\circ\eta_{2}\circ\eta_{1}\circ\eta_{2}$ |  ffffffc001ff811e719e70e70e1cec73948f1387c25e1391b0fc3 372334ad391b16874a5b12dc1b24ec96d26d269926915aa9554aa aaaaa | - | 7.07798362775
$445$ |  $\mathbb{B}_{445}^{27,13,10,7,3,3}$ |  1ffffffc001ff80e1cce3c34c6783d2663e14b913b461ec631e07 98736996a5b26c5a6f13b90fa52663d2966cf2d24cda4a955aaad 555555 | - | 7.05317709075
$446$ |  $\mathbb{B}_{445}\circ\eta_{2}$ |  3ffffff8003ff01c399c78698cf07a4cc7c29722768c3d8c63c0f 30e6d32d4b64d8b4de27721f4a4cc7a52cd9e5a499b4952ab555a aaaaaa | - | 7.12296784359
$447$ |  $\mathbb{B}_{445}\circ\eta_{2}\circ\eta_{5}$ |  7ffffff8003ff01c399c78698cf07a4cc7c29722768c3d8c63c0f 30e6d32d4b64d8b4de27721f4a4cc7a52cd9e5a499b4952ab555a aaaaaa | - | 7.19410239793
$448$ |  $\mathbb{B}_{445}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}$ |  fffffff0007fe0387338f0d319e0f4998f852e44ed187b18c781e 61cda65a96c9b169bc4ee43e94998f4a59b3cb4933692a556aab5 555554 | - | 7.168
$449$ |  $\mathbb{B}_{445}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}\circ\eta_{6}$ |  fffffff0007fe0387338f0d319e0f4998f852e44ed187b18c781e 61cda65a96c9b169bc4ee43e94998f4a59b3cb4933692a556aab5 555554 | 6.5218 | 7.1428925737
$450$ |  $\mathbb{B}_{451}\circ\eta_{4}$ |  3ffffffc000ffe01e1a70db06de07e34b5a46c93661e499938678 3733337296693998e5a6738c6e1f0f256a5c6b1cb61a5aa554aaa d555555 | - | 7.01517356059
$451$ |  $\mathbb{B}_{451}^{28,14,11,8,4}$ |  7ffffff8001ffc03c34e1b60dbc0fc696b48d926cc3c933270cf0 6e6666e52cd27331cb4ce718dc3e1e4ad4b8d6396c34b54aa9555 aaaaaaa | - | 7.17362629611
$452$ |  $\mathbb{B}_{451}\circ\eta_{2}$ |  fffffff0003ff807e1f0fc1a7c0f349b1e89ce1c6cc6693e24f0d 8726369cb4e253866cc6da4d885b18f34ad61ad4b5a56a9552aab 5555554 | - | 7.04399393187
$453$ |  $\mathbb{B}_{453}^{28,14,11,8,4}$ |  1ffffffe0007ff00fc0fcf03c66679933387859e0db07938c7987 8c79396c96996c9396b1ca59e969333996666d2b4d4ad4ab556aa a5555555 | - | 7.04991754844
$454$ |  $\mathbb{B}_{453}\circ\eta_{2}$ |  3ffffffc000ffe01f81f9e078cccf326670f0b3c1b60f2718f30f 18f272d92d32d9272d6394b3d2d266732cccda569a95a956aad55 4aaaaaaa | - | 7.04381108605
$455$ |  $\mathbb{B}_{455}^{28,14,11,8,4,4}$ |  7ffffff8001ffc03c3781e199cc785bc31ccc6696607f094f216d 8cc724cd8e172c1d2a5661e64cc934b85a4c999695a34b54aa955 5aaaaaaa | - | 7.10791045801
$456$ |  $\mathbb{B}_{455}\circ\eta_{2}$ |  fffffff0003ff80786f03c33398f0b7863998cd2cc0fe129e42db 198e499b1c2e583a54acc3cc99926970b499332d2b4696a9552aa b5555554 | - | 7.01159967629
$457$ |  $\mathbb{B}_{457}^{28,14,11,8,4,4,3}$ |  1ffffffe0007ff00f0f9cce60f3e19e4f24f6316e163391cb370b c70b670b6d0b730db9327a47b274e34e59a534a64cd94b4ab556a aa5555555 | - | 7.01966254369
$458$ |  $\mathbb{B}_{461}\circ\eta_{4}\circ\eta_{4}\circ\eta_{4}$ |  3ffffffc000ffe01e1ec66f03f834b34b1b598dce1e729cda34d8 d86691e658d8c378c9f2696c8d98393c33c35ab52e64e96956aad 554aaaaaa | - | 7.11884884273
$459$ |  $\mathbb{B}_{461}\circ\eta_{0}$ |  7ffffff0003ff80787b19bc0fe0d2cd2c6d66373879ca7368d363 619a479963630de327c9a5b23660e4f0cf0d6ad4b993a5a55aab5 552aaaaaa | - | 7.25635461872
$460$ |  $\mathbb{B}_{461}\circ\eta_{4}$ |  fffffff0003ff80787b19bc0fe0d2cd2c6d66373879ca7368d363 619a479963630de327c9a5b23660e4f0cf0d6ad4b993a5a55aab5 552aaaaaa | - | 7.39549839228
$461$ |  $\mathbb{B}_{461}^{28,14,11,8,4,4}$ |  1ffffffe0007ff00f0f633781fc1a59a58dacc6e70f394e6d1a6c 6c3348f32c6c61bc64f934b646cc1c9e19e1ad5a973274b4ab556 aaa5555555 | - | 7.53941393501
$462$ |  $\mathbb{B}_{461}\circ\eta_{1}$ |  3ffffffc000ffe01e1ec66f03f834b34b1b598dce1e729cda34d8 d86691e658d8c378c9f2696c8d98393c33c35ab52e64e96956aad 554aaaaaab | - | 7.38509445713
$463$ |  $\mathbb{B}_{461}\circ\eta_{1}\circ\eta_{1}$ |  7ffffff8001ffc03c3d8cde07f069669636b31b9c3ce539b469b1 b0cd23ccb1b186f193e4d2d91b3072786786b56a5cc9d2d2ad55a aa95555557 | - | 7.25248663644
$464$ |  $\mathbb{B}_{461}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}$ |  fffffff0003ff80787b19bc0fe0d2cd2c6d66373879ca7368d363 619a479963630de327c9a5b23660e4f0cf0d6ad4b993a5a55aab5 552aaaaaaf | - | 7.13467656416

TABLE XII: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$465$ |  $\mathbb{B}_{461}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}$ |  1ffffffe0007ff00f0f633781fc1a59a58dacc6e70f394e6d1a6c 6c3348f32c6c61bc64f934b646cc1c9e19e1ad5a973274b4ab556 aaa5555555f | - | 7.08469855832
$466$ |  $\mathbb{B}_{461}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}$ |  3ffffffc000ffe01e1ec66f03f834b34b1b598dce1e729cda34d8 d86691e658d8c378c9f2696c8d98393c33c35ab52e64e96956aad 554aaaaaabe | - | 7.09057663423
$467$ |  $\mathbb{B}_{467}^{30,15,12,10,6,6,6}$ |  7ffffffe0003ffc00fc0fc1e0c39e19ce1c725e1399933cc8db83 6691e6691e635b8dccb3199b16872496c9969b4d694ad4ad54aab 5556aaaaaaa | - | 7.0075509286
$468$ |  $\mathbb{B}_{469}\circ\eta_{4}$ |  fffffffe0003ffc03f03f19c781f870e6631e8c63ccc78f138f1c c9c96ce1c9cc92db12da4ccb64de93666d25a95a4992b52b54aab 5556aaaaaaa | - | 7.16795392067
$469$ |  $\mathbb{B}_{469}$ |  1fffffffc0007ff807e07e338f03f0e1ccc63d18c7998f1e271e3 99392d9c3939925b625b49996c9bd26ccda4b52b493256a56a955 6aaad5555555 | - | 7.3743127263
$470$ |  $\mathbb{B}_{469}\circ\eta_{1}$ |  3fffffff8000fff00fc0fc671e07e1c3998c7a318f331e3c4e3c7 32725b38727324b6c4b69332d937a4d99b496a569264ad4ad52aa d555aaaaaaab | - | 7.25165780316
$471$ |  $\mathbb{B}_{469}\circ\eta_{1}\circ\eta_{2}$ |  7fffffff0001ffe01f81f8ce3c0fc3873318f4631e663c789c78e 64e4b670e4e6496d896d2665b26f49b33692d4ad24c95a95aa555 aaab55555556 | - | 7.12719270064
$472$ |  $\mathbb{B}_{469}\circ\eta_{1}\circ\eta_{2}\circ\eta_{2}$ |  fffffffe0003ffc03f03f19c781f870e6631e8c63ccc78f138f1c c9c96ce1c9cc92db12da4ccb64de93666d25a95a4992b52b54aab 5556aaaaaaac | - | 7.07340614681
$473$ |  $\mathbb{B}_{473}^{29,15,11,9,6,6,4}$ |  1fffffff0001ffc01f81f2661c3ec1b324ce61e3c363633c396cd 0edc2e4e42dc4bcc792d327272d25a64ce331ac52da6635a95aad 55aaab5555555 | - | 7.38087226181
$474$ |  $\mathbb{B}_{473}\circ\eta_{2}$ |  3ffffffe0003ff803f03e4cc387d8366499cc3c786c6c67872d9a 1db85c9c85b89798f25a64e4e5a4b4c99c66358a5b4cc6b52b55a ab5556aaaaaaa | - | 7.31129189717
$475$ |  $\mathbb{B}_{473}\circ\eta_{2}\circ\eta_{5}$ |  7ffffffe0003ff803f03e4cc387d8366499cc3c786c6c67872d9a 1db85c9c85b89798f25a64e4e5a4b4c99c66358a5b4cc6b52b55a ab5556aaaaaaa | - | 7.24410839273
$476$ |  $\mathbb{B}_{473}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}$ |  fffffffc0007ff007e07c99870fb06cc9339878f0d8d8cf0e5b34 3b70b9390b712f31e4b4c9c9cb49699338cc6b14b6998d6a56ab5 56aaad5555554 | - | 7.07430997877
$477$ |  $\mathbb{B}_{477}^{29,15,11,9,6,6,4}$ |  1fffffff0001ffc01f81e6f03c78c3e61e463e0f3333cccb4cb78 c67878c2c96966c970cf0ccd33334a526e5a652c96d2b465a95aa d55aaab5555555 | - | 7.27859884837
$478$ |  $\mathbb{B}_{477}\circ\eta_{2}$ |  3ffffffe0003ff803f03cde078f187cc3c8c7c1e66679996996f1 8cf0f18592d2cd92e19e199a666694a4dcb4ca592da568cb52b55 aab5556aaaaaaa | - | 7.18548336373
$479$ |  $\mathbb{B}_{479}^{29,15,11,9,6,6,4}$ |  7ffffffc0007ff007e07e72331c993c61b619d25ccb0f03c4b31b 0c3c1b4c394b4d3933c4b52d3cc8709963964b19c933726a56a55 2aa5554aaaaaaa | - | 7.12063186643
$480$ |  $\mathbb{B}_{479}\circ\eta_{1}$ |  fffffff8000ffe00fc0fce466393278c36c33a4b9961e07896636 1878369872969a7267896a5a7990e132c72c96339266e4d4ad4aa 554aaa95555555 | - | 7.05882352941
$481$ |  $\mathbb{B}_{481}^{29,15,11,9,6,6,4}$ |  1fffffff0001ffc01f81fc27e1b6cc92f0cda6627a52f2d33b273 9a49e2d33c258e19363133c343e162661ccb438cc71a562d5a95a ad55aaab5555555 | - | 7.26636306533
$482$ |  $\mathbb{B}_{481}\circ\eta_{1}$ |  3ffffffe0003ff803f03f84fc36d9925e19b4cc4f4a5e5a6764e7 3493c5a6784b1c326c62678687c2c4cc399687198e34ac5ab52b5 5aab5556aaaaaab | - | 7.15944530046
$483$ |  $\mathbb{B}_{481}\circ\eta_{1}\circ\eta_{2}$ |  7ffffffc0007ff007e07f09f86db324bc3369989e94bcb4cec9ce 69278b4cf0963864d8c4cf0d0f858998732d0e331c6958b56a56a b556aaad5555556 | - | 7.17503229378
$484$ |  $\mathbb{B}_{481}\circ\eta_{1}\circ\eta_{2}\circ\eta_{6}$ |  7ffffffc0007ff007e07f09f86db324bc3369989e94bcb4cec9ce 69278b4cf0963864d8c4cf0d0f858998732d0e331c6958b56a56a b556aaad5555556 | - | 7.08406919076
$485$ |  $\mathbb{B}_{481}\circ\eta_{1}\circ\eta_{2}\circ\eta_{6}\circ\eta_{6}$ |  7ffffffc0007ff007e07f09f86db324bc3369989e94bcb4cec9ce 69278b4cf0963864d8c4cf0d0f858998732d0e331c6958b56a56a b556aaad5555556 | - | 7.11165195308

TABLE XIII: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$486$ |  $\mathbb{B}_{487}\circ\eta_{4}$ |  3fffffff0007ff807e07cf038f8c79c1e60f8799b0dc9f30f3391 b0e730f48f4b364b1b9334b358dcb199694a65ad96c9492b4d6a5 6a9556aab5555555 | - | 7.14230420321
$487$ |  $\mathbb{B}_{487}^{30,13,12,8,6,6,5}$ |  7ffffffe000fff00fc0f9e071f18f383cc1f0f3361b93e61e6723 61ce61e91e966c9637266966b1b96332d294cb5b2d9292569ad4a d52aad556aaaaaaa | - | 7.19784522003
$488$ |  $\mathbb{B}_{487}\circ\eta_{1}$ |  fffffffc001ffe01f81f3c0e3e31e707983e1e66c3727cc3cce46 c39cc3d23d2cd92c6e4cd2cd6372c665a52996b65b2524ad35a95 aa555aaad5555555 | - | 7.19033816425
$489$ |  $\mathbb{B}_{489}^{30,15,12,8,6,4}$ |  1fffffff8000fff00fc2c7e07878e1ec1f8c6d927c99626679339 86d24f264e634e3c6993396662798d639c6c95ac5a49696a56c2d 4ab554aaa95555555 | - | 7.20070464948
$490$ |  $\mathbb{B}_{489}\circ\eta_{2}$ |  3fffffff0001ffe01f858fc0f0f1c3d83f18db24f932c4ccf2673 0da49e4c9cc69c78d32672ccc4f31ac738d92b58b492d2d4ad85a 956aa95552aaaaaaa | - | 7.09305760709
$491$ |  $\mathbb{B}_{489}\circ\eta_{2}\circ\eta_{1}$ |  7ffffffe0003ffc03f0b1f81e1e387b07e31b649f2658999e4ce6 1b493c99398d38f1a64ce59989e6358e71b256b16925a5a95b0b5 2ad552aaa55555555 | - | 7.08519955328
$492$ |  $\mathbb{B}_{495}\circ\eta_{0}\circ\eta_{4}$ |  fffffff0001ffc01f81f06d836cc2fc30d9c3b659b4e4ed38d8da 4e72666961e66726c78d8db0ec6c39863b498d34af4ce358e5295 a954aa95552aaaaaa | - | 7.07045215563
$493$ |  $\mathbb{B}_{495}\circ\eta_{4}\circ\eta_{4}$ |  1fffffff0001ffc01f81f06d836cc2fc30d9c3b659b4e4ed38d8d a4e72666961e66726c78d8db0ec6c39863b498d34af4ce358e529 5a954aa95552aaaaaa | - | 7.0810220254
$494$ |  $\mathbb{B}_{495}\circ\eta_{1}\circ\eta_{3}\circ\eta_{3}$ |  3ffffff8000ffe00fc0f836c1b6617e186ce1db2cda72769c6c6d 27393334b0f3339363c6c6d876361cc31da4c69a57a671ac7294a d4aa554aaa95555555 | - | 7.07637882039
$495$ |  $\mathbb{B}_{495}^{29,15,11,9,6,6,4}$ |  7ffffffc0007ff007e07c1b60db30bf0c3670ed966d393b4e3636 939c999a587999c9b1e3636c3b1b0e618ed2634d2bd338d6394a5 6a552aa5554aaaaaaa | - | 7.01233472612
$496$ |  $\mathbb{B}_{495}\circ\eta_{1}$ |  fffffff8000ffe00fc0f836c1b6617e186ce1db2cda72769c6c6d 27393334b0f3339363c6c6d876361cc31da4c69a57a671ac7294a d4aa554aaa95555555 | - | 7.01459854015
$497$ |  $\mathbb{B}_{495}\circ\eta_{1}\circ\eta_{5}$ |  1fffffff8000ffe00fc0f836c1b6617e186ce1db2cda72769c6c6 d27393334b0f3339363c6c6d876361cc31da4c69a57a671ac7294 ad4aa554aaa95555555 | - | 7.01730113636
$498$ |  $\mathbb{B}_{497}^{27,15,11,11,7,7,7,7}\circ\eta_{2}$ |  3ffffff8000ffe003f80fe03c9c86321b0e61a7387983ccb724e6 696c3d26636670b4e1e66c723ccb59a5b27966d397365c9cb56ad 5ab556aad555aaaaaaa | - | 7.00378424174
$499$ |  $\mathbb{B}_{499}^{29,15,11,9,6,6,4}$ |  7ffffffc0007ff007e07d87431ed839c0e4f86ce46978d8f49b67 2d9b19b4c9cc399398f2639c2d8da1e46ce5ac6d49b58e934258a 56a552aa5554aaaaaaa | - | 7.16797167367
$500$ |  $\mathbb{B}_{499}\circ\eta_{1}$ |  fffffff8000ffe00fc0fb0e863db07381c9f0d9c8d2f1b1e936ce 5b3633699398732731e4c7385b1b43c8d9cb58da936b1d2684b14 ad4aa554aaa95555555 | - | 7.04542892571
$501$ |  $\mathbb{B}_{501}^{30,15,12,8,6,4}$ |  1fffffff8000fff00fc1e4f30f865f207e0f34da63598c5e0d939 bc360e4d8ccc9ce4a72d1939ca5ec99f261cf34a56a35e694b34e 5ad4ab554aaa95555555 | - | 7.16572456321
$502$ |  $\mathbb{B}_{501}\circ\eta_{2}$ |  3fffffff0001ffe01f83c9e61f0cbe40fc1e69b4c6b318bc1b273 786c1c9b199939c94e5a327394bd933e4c39e694ad46bcd29669c b5a956aa95552aaaaaaa | - | 7.27872451043
$503$ |  $\mathbb{B}_{501}\circ\eta_{2}\circ\eta_{5}$ |  7fffffff0001ffe01f83c9e61f0cbe40fc1e69b4c6b318bc1b273 786c1c9b199939c94e5a327394bd933e4c39e694ad46bcd29669c b5a956aa95552aaaaaaa | - | 7.39489682586
$504$ |  $\mathbb{B}_{501}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}$ |  fffffffe0003ffc03f0793cc3e197c81f83cd3698d663178364e6 f0d83936333273929cb464e7297b267c9873cd295a8d79a52cd39 6b52ad552aaa55555554 | - | 7.21964529332
$505$ |  $\mathbb{B}_{501}\circ\eta_{2}\circ\eta_{2}\circ\eta_{1}\circ\eta_{5}$ |  1fffffffc0007ff807e0f27987c32f903f079a6d31acc62f06c9c de1b0726c6664e7253968c9ce52f64cf930e79a52b51af34a59a7 2d6a55aaa5554aaaaaaa9 | - | 7.09980512249

TABLE XIV: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$506$ |  $\mathbb{B}_{501}\circ\eta_{2}\circ\eta_{2}\circ\eta_{1}\circ\eta_{1}\circ\eta_{5}$ |  3fffffff8000fff00fc1e4f30f865f207e0f34da63598c5e0d939 bc360e4d8ccc9ce4a72d1939ca5ec99f261cf34a56a35e694b34e 5ad4ab554aaa955555553 | - | 7.05994595489
$507$ |  $\mathbb{B}_{507}^{29,15,11,9,6,6,4}$ |  7ffffffc0007ff007e07c1b30db303f0e1c9c6d99996e58f4f0e7 863948cf183592cdc1b65a6d2c2d86e19998e49c96d2b5338d339 4a56a552aa5554aaaaaaa | - | 7.23144657627
$508$ |  $\mathbb{B}_{507}\circ\eta_{1}$ |  fffffff8000ffe00fc0f83661b6607e1c3938db3332dcb1e9e1cf 0c72919e306b259b836cb4da585b0dc33331c9392da56a671a672 94ad4aa554aaa95555555 | - | 7.10293955741
$509$ |  $\mathbb{B}_{509}^{26,13,12,9,7,6,6,4,2}$ |  1ffffff8003ffc01fc0fc3078f266781f81f393278cda4cb1c78d 2649c6963ccccd2786d8e63c96db0ce1cc9633935a95a96663496 b2d4ad5aad552aa9555555 | - | 7.03489193005
$510$ |  $\mathbb{B}_{511}\circ\eta_{4}$ |  3ffffffe0007ffc007f01fc079b43c5b0ce72761e437872d2cccb 4e4db61b18ccc9b1a71ce4f0ccc3c36972e5a76364cb1ed2f196a d5ab56aad556aaa5555555 | - | 7.00889248181
$511$ |  $\mathbb{B}_{511}^{27,15,11,11,7,7,7,7}$ |  7ffffff0001ffc007f01fc079258d264c3613c61b1b1c8d8665a5 8c78c6c4b4b3c3c4e4da4d8786658dc9393964b1634c670d871a5 4a952a554aa95552aaaaaa | - | 7.22166602135
$512$ |  $\mathbb{B}_{511}\circ\eta_{1}$ |  ffffffe0003ff800fe03f80f24b1a4c986c278c3636391b0ccb4b 18f18d8969678789c9b49b0f0ccb1b927272c962c698ce1b0e34a 952a54aa9552aaa5555555 | - | 7.09417622862
$513$ |  $\mathbb{B}_{513}^{29,14,13,10,7,7,7,7,2}$ |  1fffffff0003ffe007f01fc078c6f0cc799c9e1b36b4e3ca52d61 c7a49ce66679966664d8e16da7c3e0d24f0731a58d996ccb46c96 ad5ab56aa5552aab5555555 | - | 7.11729229771
$514$ |  $\mathbb{B}_{513}\circ\eta_{2}$ |  3ffffffe0007ffc00fe03f80f18de198f3393c366d69c794a5ac3 8f4939ccccf32cccc9b1c2db4f87c1a49e0e634b1b32d9968d92d 5ab56ad54aaa5556aaaaaaa | - | 7.02014136153
$515$ |  $\mathbb{B}_{517}\circ\eta_{0}$ |  7fffffff0001ffe007e07c3c1d83e493b670b658c79c3a5c3cdb1 32c66c70f263672d24e64f3138cb487b49a4d863d263b1c6b5894 b4a56a556aa95552aaaaaaa | - | 7.00874689498
$516$ |  $\mathbb{B}_{517}\circ\eta_{4}$ |  ffffffff0001ffe007e07c3c1d83e493b670b658c79c3a5c3cdb1 32c66c70f263672d24e64f3138cb487b49a4d863d263b1c6b5894 b4a56a556aa95552aaaaaaa | - | 7.01190350785
$517$ |  $\mathbb{B}_{517}^{27,15,11,11,7,7,7,7}$ |  1ffffffc0007ff001fc07f01e1e0f0f270cf03e19cc3c664d86d8 7c63932d999ccd999c33926d69c69ce66d2cd9a52b4cb634b4a5a 5ab56ad5aab556aaad555555 | - | 7.12922756855
$518$ |  $\mathbb{B}_{517}\circ\eta_{2}$ |  3ffffff8000ffe003f80fe03c3c1e1e4e19e07c339878cc9b0db0 f8c7265b33399b33386724dad38d39ccda59b34a56996c69694b4 b56ad5ab556aad555aaaaaaa | - | 7.20719849584
$519$ |  $\mathbb{B}_{519}^{27,15,11,11,7,7,7,7}$ |  7ffffff0001ffc007f01fc07c36493e1b64d25b0f87cc793ccf98 da7271b0f1999992d3927278d9accb1a4ca5ad3870c6396b1c634 a54a952a554aa95552aaaaa | - | 7.18717647687
$520$ |  $\mathbb{B}_{519}\circ\eta_{1}$ |  ffffffe0003ff800fe03f80f86c927c36c9a4b61f0f98f2799f31 b4e4e361e3333325a724e4f1b3599634994b5a70e18c72d638c69 4a952a54aa9552aaa555555 | - | 7.06078963860
$521$ |  $\mathbb{B}_{521}^{27,15,11,11,7,7,7,7}$ |  1ffffffc0007ff001fc07f01b2c4996c7c36d0f60f358f0e63639 c9d878cce1e4f4e5a4cc969d8d927264b49f34a74bc72d6c798ec 31ab56ad5aab556aaad55555 | - | 7.01760599793
$522$ |  $\mathbb{B}_{521}\circ\eta_{2}$ |  3ffffff8000ffe003f80fe03658932d8f86da1ec1e6b1e1cc6c73 93b0f199c3c9e9cb49992d3b1b24e4c9693e694e978e5ad8f31d8 6356ad5ab556aad555aaaaaa | - | 7.06759350521
$523$ |  $\mathbb{B}_{521}\circ\eta_{2}\circ\eta_{5}$ |  7ffffff8000ffe003f80fe03658932d8f86da1ec1e6b1e1cc6c73 93b0f199c3c9e9cb49992d3b1b24e4c9693e694e978e5ad8f31d8 6356ad5ab556aad555aaaaaa | - | 7.11833133816
$524$ |  $\mathbb{B}_{521}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}$ |  fffffff0001ffc007f01fc06cb1265b1f0db43d83cd63c398d8e7 2761e3338793d39693325a763649c992d27cd29d2f1cb5b1e63b0 c6ad5ab56aad55aaab555555 | - | 7.08181161663
$525$ |  $\mathbb{B}_{521}\circ\eta_{2}\circ\eta_{2}\circ\eta_{5}\circ\eta_{6}$ |  fffffff0001ffc007f01fc06cb1265b1f0db43d83cd63c398d8e7 2761e3338793d39693325a763649c992d27cd29d2f1cb5b1e63b0 c6ad5ab56aad55aaab555555 | - | 7.04634931997
$526$ |  $\mathbb{B}_{521}\circ\eta_{2}\circ\eta_{2}\circ\eta_{1}\circ\eta_{1}\circ\eta_{5}$ |  3ffffffc0007ff001fc07f01b2c4996c7c36d0f60f358f0e63639 c9d878cce1e4f4e5a4cc969d8d927264b49f34a74bc72d6c798ec 31ab56ad5aab556aaad555555 | - | 7.03401637260
$527$ |  $\mathbb{B}_{517}\circ\eta_{2}\circ\eta_{2}\circ\eta_{1}\circ\eta_{1}\circ\eta_{2}\circ\eta_{5}\circ\eta_{6}\circ\eta_{6}\circ\eta_{5}\circ\eta_{5}$ |  67ffffff8000ffe003f80fe03c3c1e1e4e19e07c339878cc9b0db 0f8c7265b33399b33386724dad38d39ccda59b34a56996c69694b 4b56ad5ab556aad555aaaaaaa6 | - | 7.08239404294

TABLE XV: A list of binary sequences with record merit factor values $n$ |  Class |  Record sequence in HEX | Old MF | New MF
---|---|---|---|---
$573$ |  $\mathbb{B}_{573}^{27,15,11,11,7,7,7,7}$ |  1ffffffc0007ff001fc07f01e7c2787b0e1e5a7e3cf0f64bc39ce 372d9b399ac9b26e333333246318c199319c3724d92d0e74b4d25 61e5a4b16962d65ab56ad5aab556aaad55555 | - | 6.82937432399
$1006$ |  $\mathbb{B}_{1009}\circ\eta_{0}\circ\eta_{4}$ |  3fffffffff000007fff0001ff801fe01e03fc0be03e9c60fb70e1 f039b0e3c762c73ca479ccc6f0d9c9327c97879938d8dcd937330 e1793867d8761e19f1b59f1b59a5a769d669397a4b33739cdc9c9 3996978d6338d9cb46ccd96e0d36c276d24b192b5a4b714a6d852 a50ad52a5aa55aa955aaab5556aaaab55555555 | - | 6.35677047348
$1007$ |  $\mathbb{B}_{1009}\circ\eta_{0}$ |  7ffffffffe00000fffe0003ff003fc03c07f817c07d38c1f6e1c3 e07361c78ec58e7948f3998de1b39264f92f0f3271b1b9b26e661 c2f270cfb0ec3c33e36b3e36b34b4ed3acd272f49666e739b9392 732d2f1ac671b3968d99b2dc1a6d84eda4963256b496e294db0a5 4a15aa54b54ab552ab5556aaad55556aaaaaaaa | - | 6.41941303825
$1008$ |  $\mathbb{B}_{1009}\circ\eta_{4}$ |  fffffffffe00000fffe0003ff003fc03c07f817c07d38c1f6e1c3 e07361c78ec58e7948f3998de1b39264f92f0f3271b1b9b26e661 c2f270cfb0ec3c33e36b3e36b34b4ed3acd272f49666e739b9392 732d2f1ac671b3968d99b2dc1a6d84eda4963256b496e294db0a5 4a15aa54b54ab552ab5556aaad55556aaaaaaaa | - | 6.41811107180
$1009$ |  $\mathbb{B}_{1009}^{39,21,15,15,10,10,8,8,4}$ |  1fffffffffc00001fffc0007fe007f80780ff02f80fa7183edc38 7c0e6c38f1d8b1cf291e7331bc36724c9f25e1e64e3637364dccc 385e4e19f61d87867c6d67c6d66969da759a4e5e92ccdce737272 4e65a5e358ce3672d1b3365b834db09db492c64ad692dc529b614 a942b54a96a956aa556aaad555aaaaad55555555 | - | 6.41690827959
$1010$ |  $\mathbb{B}_{1009}\circ\eta_{2}$ |  3fffffffff800003fff8000ffc00ff00f01fe05f01f4e307db870 f81cd871e3b1639e523ce663786ce4993e4bc3cc9c6c6e6c9b998 70bc9c33ec3b0f0cf8dacf8dacd2d3b4eb349cbd2599b9ce6e4e4 9ccb4bc6b19c6ce5a3666cb7069b613b69258c95ad25b8a536c29 52856a952d52ad54aad555aaab55555aaaaaaaaa | - | 6.36726796080