# Hadamard matrices of order $36$ formed by codewords in some ternary self-
dual codes

Masaaki Harada and Keita Ishizuka  Research Center for Pure and Applied
Mathematics, Graduate School of Information Sciences, Tohoku University,
Sendai 980–8579, Japan. email<EMAIL_ADDRESS>Research Center for Pure
and Applied Mathematics, Graduate School of Information Sciences, Tohoku
University, Sendai 980–8579, Japan. email<EMAIL_ADDRESS>
###### Abstract

In this note, we study the existence of Hadamard matrices of order $36$ formed
by codewords of weight $36$ in some ternary near-extremal self-dual codes of
length $36$.

Keywords: Hadamard matrix, ternary self-dual code, four-negacirculant code

##  1 Introduction

Self-dual codes are one of the most interesting classes of codes. This
interest is justified by many combinatorial objects and algebraic objects
related to self-dual codes (see e.g., [13]). Among these objects are Hadamard
matrices. In particular, a special class of Hadamard matrices can give rise to
self-dual codes as their row spaces. In this note, we investigate the
existence of Hadamard matrices of order $36$ formed by codewords of weight
$36$ in some ternary near-extremal self-dual codes of length $36$.

A ternary self-dual code $C$ of length $n$ is an $[n,n/2]$ code over the
finite field of order $3$ satisfying $C=C^{\perp}$, where $C^{\perp}$ is the
dual code of $C$. A ternary self-dual code of length $n$ exists if and only if
$n$ is divisible by four. It was shown in [10] that the minimum weight $d$ of
a ternary self-dual code of length $n$ is bounded by $d\leq 3\lfloor
n/12\rfloor+3$. If $d=3\lfloor n/12\rfloor+3$ (resp. $d=3\lfloor
n/12\rfloor$), then the code is called extremal (resp. near-extremal). For
length $36$, the Pless symmetry code is a currently known extremal self-dual
code.

Recently, Tonchev [17] studied Hadamard matrices of order $n$ formed by
codewords of weight $n$ in ternary extremal self-dual codes of length $n$. The
Pless symmetry code contains a Hadamard matrix [11, Theorem 4.2]. This matrix
is often called the Paley-Hadamard matrix of type II (see [17]). Tonchev [17]
showed that the Pless symmetry code of length $36$ contains exactly two
inequivalent Hadamard matrices of order $36$, namely, the Paley-Hadamard
matrix of type II and a regular Hadamard matrix. Inspired by his result, we
study the existence of Hadamard matrices of order $36$ formed by codewords of
weight $36$ in some ternary near-extremal self-dual codes of length $36$.

This note is organized as follows. In Section 2, we give definitions and some
known results of ternary self-dual codes and Hadamard matrices. In Section 3,
we investigate ternary self-dual codes generated by the rows of Hadamard
matrices of order $36$ with automorphism of order $17$ classified in [16]. We
also give a classification of ternary near-extremal bordered double circulant
self-dual codes, four-negacirculant self-dual codes and quasi-twisted self-
dual codes of length $36$. In Section 4, we study the existence of Hadamard
matrices formed by codewords of weight $36$ in the ternary near-extremal self-
dual codes of length $36$ classified in Section 3. Consequently, $89$
inequivalent Hadamard matrices of order $36$ are constructed from codewords of
weight $36$ in the ternary near-extremal self-dual codes. We also investigate
ternary codes generated by the rows of the $89$ Hadamard matrices and their
transposes. Finally, we give a relationship between the Pless symmetry code
and a certain near-extremal four-negacirculant self-dual code via the regular
Hadamard matrix given in [17, Theorem 2.4 (i)].

##  2 Preliminaries

In this section, we give definitions and some known results of ternary self-
dual codes and Hadamard matrices used in this note.

###  2.1 Ternary self-dual codes

Let $\mathbb{F}_{3}=\\{0,1,2\\}$ denote the finite field of order $3$. A
_ternary_ $[n,k]$ _code_ $C$ is a $k$-dimensional vector subspace of
$\mathbb{F}_{3}^{n}$. All codes in this note are ternary, and the expression
“codes” means “ternary codes”. The parameters $n$ and $k$ are called the
_length_ and _dimension_ of $C$, respectively. A _generator matrix_ of $C$ is
a $k\times n$ matrix whose rows are a basis of $C$. The _weight_
$\operatorname{wt}(x)$ of a vector $x$ of $\mathbb{F}_{3}^{n}$ is the number
of non-zero components of $x$. A vector of $C$ is called a _codeword_. The
minimum non-zero weight of all codewords in $C$ is called the _minimum weight_
of $C$. The polynomial $\sum_{c\in C}y^{\operatorname{wt}(c)}$ is called the
_weight enumerator_ of $C$.

The dual code $C^{\perp}$ of a code $C$ of length $n$ is defined as
$C^{\perp}=\\{x\in\mathbb{F}_{3}^{n}\mid x\cdot y=0\text{ for all }y\in C\\},$
where $x\cdot y$ is the standard inner product. A code $C$ is _self-dual_ if
$C=C^{\perp}$. A self-dual code of length $n$ exists if and only if $n$ is
divisible by four. Two codes $C$ and $C^{\prime}$ are _equivalent_ if there is
a monomial matrix $P$ over $\mathbb{F}_{3}$ with $C^{\prime}=C\cdot P$, where
$C\cdot P=\\{xP\mid x\in C\\}$. We denote two equivalent codes $C$ and $D$ by
$C\cong D$. All self-dual codes were classified in [3], [5], [9] and [12] for
lengths up to $24$.

By the Gleason theorem (see [10]), the weight enumerator of an extremal self-
dual code is uniquely determined for each length. Moreover, it was shown in
[10] that the minimum weight $d$ of a self-dual code of length $n$ is bounded
by $d\leq 3\lfloor n/12\rfloor+3$. If $d=3\lfloor n/12\rfloor+3$, then the
code is called _extremal_. For $n\in\\{4,8,12,\ldots,64\\}$, it is known that
there is an extremal self-dual code of length $n$ (see [6, Table 6]). For
length $36$, the Pless symmetry code $P_{36}$ (see Section 2.3 for the
definition) is a currently known extremal self-dual code. If $d=3\lfloor
n/12\rfloor$, then the code is called _near-extremal_. Recently, some
restrictions on the weight enumerators of near-extremal self-dual codes of
length divisible by $12$ are given in [1]. For example, the possible weight
enumerators of near-extremal self-dual codes of length $36$ are as follows:

$\begin{split}&1+\alpha
y^{9}+(42840-9\alpha)y^{12}+(1400256+36\alpha)y^{15}\\\
&+(18452280-84\alpha)y^{18}+(90370368+126\alpha)y^{21}\\\
&+(162663480-126\alpha)y^{24}+(97808480+84\alpha)y^{27}\\\
&+(16210656-36\alpha)y^{30}+(471240+9\alpha)y^{33}+(888-\alpha)y^{36},\end{split}$
(1)

where $\alpha=8\beta$ and $\beta\in\\{1,2,\ldots,111\\}$ [1]. Note that the
number of codewords of weight $9$ in a near-extremal self-dual code $C$ of
length $36$ determines the weight enumerator of $C$.

###  2.2 Hadamard matrices

A _Hadamard_ matrix $H$ of order $n$ is an $n\times n$ matrix whose entries
are from $\\{1,-1\\}$ such that $HH^{T}=nI_{n}$, where $H^{T}$ is the
transpose of $H$ and $I_{n}$ is the identity matrix of order $n$. Hadamard
matrices are widely investigated combinatorial matrices (see e.g., [15]).
Hadamard matrices are also one of combinatorial matrices related to self-dual
codes.

It is known that the order $n$ is necessarily $1,2$, or a multiple of $4$. Two
Hadamard matrices $H$ and $K$ are said to be _equivalent_ if there are
$(1,-1,0)$-monomial matrices $P$ and $Q$ with $K=PHQ$. We denote two
equivalent Hadamard matrices $H$ and $K$ by $H\cong K$. An _automorphism_ of a
Hadamard matrix $H$ is an equivalence of $H$ to itself. The set of all
automorphisms of $H$ forms a group under composition called the _automorphism
group_ $\operatorname{Aut}(H)$ of $H$. For orders up to $32$, all inequivalent
Hadamard matrices are known (see [8] and [7] for orders $28$ and $32$,
respectively). Many Hadamard matrices of order $36$ are known (see e.g.,
[14]). Tonchev [16] showed that there are $11$ inequivalent Hadamard matrices
of order $36$ with automorphism of order $17$. Note that one of them is the
well-known Paley-Hadamard matrix of type II (see [17] for the definition).

###  2.3 Pless symmetry code $P_{36}$ of length $36$

An $n\times n$ matrix of the following form

$\left(\begin{array}[]{cccccc}r_{0}&r_{1}&r_{2}&\cdots&r_{n-2}&r_{n-1}\\\ \mu
r_{n-1}&r_{0}&r_{1}&\cdots&r_{n-3}&r_{n-2}\\\ \mu r_{n-2}&\mu
r_{n-1}&r_{0}&\cdots&r_{n-4}&r_{n-3}\\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\\ \mu r_{1}&\mu r_{2}&\mu
r_{3}&\cdots&\mu r_{n-1}&r_{0}\end{array}\right)$

is called _circulant_ and _negacirculant_ if $\mu=1$ and $-1$, respectively.
If $R$ is an $(n-1)\times(n-1)$ circulant matrix, then a $[2n,n]$ code with
generator matrix of the following form

$\left(\begin{array}[]{ccccccccc}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&0&1&\cdots&1\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&1&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&I_{n}&{}\hfil&{}\hfil&\vdots&{}\hfil&R&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&1&{}\hfil&{}\hfil&{}\hfil\\\
\end{array}\right)$ (2)

is called a _bordered double circulant_ code. The Pless symmetry code $P_{36}$
is defined as the bordered double circulant code with generator matrix of the
form (2), where

$(0,1,1,2,1,2,2,2,1,1,2,2,2,1,2,1,1)$

is the first row of $R$ [11].

Now we give other construction methods of the Pless symmetry code $P_{36}$. If
$N$ is an $n\times n$ negacirculant matrix, then a $[2n,n]$ code with
generator matrix of the following form

$\left(\begin{array}[]{cc}I_{n}&N\end{array}\right)$ (3)

is called a _quasi-twisted_ code. There is a quasi-twisted self-dual code $C$
such that $C\cong P_{36}$ [4]. If $A$ and $B$ are $n\times n$ negacirculant
matrices, then a $[4n,2n]$ code with generator matrix of the following form

$\left(\begin{array}[]{ccc@{}c}&{I_{2n}}&&\begin{array}[]{cc}A&B\\\
2B^{T}&A^{T}\end{array}\end{array}\right)$ (4)

is called a _four-negacirculant_ code. There is a four-negacirculant self-dual
code $C$ such that $C\cong P_{36}$ [4].

Let $H$ be a Hadamard matrix of order $n$. Throughout this note, we denote by
$C(H)$ the code generated by the rows of $H$, where entries of $H$ are
regarded as elements of $\mathbb{F}_{3}$. As described above, Tonchev [16]
showed that there are $11$ inequivalent Hadamard matrices of order $36$ with
automorphism of order $17$. We denote the Hadamard matrix corresponding to the
$3$-$(36,18,8)$ design $E_{i}$ in [16, Table 4] by $T_{36,i}$
$(i=1,2,\ldots,11)$. Note that $T_{36,11}$ is equivalent to the Paley-Hadamard
matrix of type II. This implies that $C(T_{36,11})\cong P_{36}$.

Here we regard the Pless symmetry code $P_{36}$ as a member of the following
four families of codes:

  * (F1)

codes $C(T_{36,i})$ from the Hadamard matrices $T_{36,i}$ $(i=1,2,\ldots,11)$,

  * (F2)

bordered double circulant self-dual codes,

  * (F3)

four-negacirculant self-dual codes,

  * (F4)

quasi-twisted self-dual codes.

##  3 Near-extremal self-dual codes of length $36$

For the above four families (F1)–(F4), we consider near-extremal self-dual
codes of length $36$. All computer calculations in the rest of this note were
done with the help of Magma [2].

###  3.1 Self-dual codes $C(T_{36,i})$ from $T_{36,i}$

We verified that $C(T_{36,i})$ are self-dual codes of minimum weight $9$
$(i=1,2,\ldots,10)$. In addition, we verified that $C(T_{36,2})\cong
C(T_{36,9})$ and there is no other pair of equivalent codes among
$C(T_{36,i})$. The numbers $A_{9}$ of codewords of weight $9$ in $C(T_{36,i})$
are listed in Table 1. These numbers determine the weight enumerators of
$C(T_{36,i})$ (see (1)).

Table 1: Numbers $A_{9}$ of codewords of weight $9$ in $C(T_{36,i})$
---
$A_{9}$ | $i$
136 | $1,3,5,6,7,8,10$
544 | $2,4,9$

###  3.2 Bordered double circulant self-dual codes

By exhaustive search, we found all distinct near-extremal bordered double
circulant self-dual codes of length $36$, which must be checked further for
equivalences. By equivalence testing, we verified that there are exactly $12$
inequivalent near-extremal bordered double circulant self-dual codes. We
denote these codes by $D_{36,i}$ ($i=1,2,\ldots,12$). For the codes
$D_{36,i}$, the first rows $r$ of the circulant matrices $R$ in their
generator matrices (2) are listed in Table 2. The numbers $A_{9}$ of codewords
of weight $9$ in $D_{36,i}$ are listed in Table 3. These numbers determine the
weight enumerators of $D_{36,i}$ (see (1)).

Table 2: Bordered double circulant codes $D_{36,i}$
---
$i$ | $r$ | $i$ | $r$ | $i$ | $r$
1 | 21022212010000011 | 5 | 10210211100100011 | 9 | 01111221220100112
2 | 11122112102100021 | 6 | 20211112121100012 | 10 | 20111111001000012
3 | 12121111001000010 | 7 | 12121120112100012 | 11 | 10011210000000021
4 | 02212101201000012 | 8 | 12222022210100121 | 12 | 01212112121100012
Table 3: Numbers $A_{9}$ of codewords of weight $9$ in $D_{36,i}$
---
$A_{9}$ | $i$
136 | $1,2,\ldots,7$
408 | $8,9,10$
544 | $11,12$

###  3.3 Four-negacirculant self-dual codes

By exhaustive search, we found all distinct near-extremal four-negacirculant
self-dual codes of length $36$, which must be checked further for
equivalences. By equivalence testing, we verified that there are exactly $260$
inequivalent near-extremal four-negacirculant self-dual codes. We denote these
codes by $F_{36,i}$ ($i=1,2,\ldots,260$). For the codes $F_{36,i}$, the first
rows $r_{A}$ (resp. $r_{B}$) of the negacirculant matrices $A$ (resp. $B$) in
their generator matrices (4) are listed in Tables 7 and 8. The numbers $A_{9}$
of codewords of weight $9$ in $F_{36,i}$ are listed in Table 4. These numbers
determine the weight enumerators of $F_{36,i}$ (see (1)).

Table 4: Numbers $A_{9}$ of codewords of weight $9$ in $F_{36,i}$
---
$A_{9}$ | $i$ | $A_{9}$ | $i$ | $A_{9}$ | $i$
72 | $1,2,\ldots,6$ | 312 | $105,106,\ldots,121$ | 576 | $246,247,248,249$
96 | $7,8$ | 360 | $122,123,\ldots,154$ | 600 | $250,251$
144 | $9,10,\ldots,22$ | 384 | $155,156,\ldots,171$ | 648 | $252,253,254,255$
168 | $23,24,25,26$ | 432 | $172,173,\ldots,196$ | 720 | $256,257,258$
216 | $27,28,\ldots,45$ | 456 | $197,198,\ldots,213$ | 744 | $259,260$
240 | $46,47,\ldots,60$ | 504 | $214,215,\ldots,237$ |  |
288 | $61,62,\ldots,104$ | 528 | $238,239,\ldots,245$ |  |

###  3.4 Quasi-twisted self-dual codes

By exhaustive search, we found all distinct near-extremal quasi-twisted self-
dual codes of length $36$, which must be checked further for equivalences. By
equivalence testing, we verified that there are exactly $55$ inequivalent
near-extremal quasi-twisted self-dual codes. We denote these codes by
$Q_{36,i}$ ($i=1,2,\ldots,55$).

###  3.5 Equivalent codes

We investigate equivalences among the near-extremal self-dual codes
$C(T_{36,i})$, $D_{36,i}$, $F_{36,i}$ and $Q_{36,i}$. We verified that

$\begin{array}[]{lll}D_{36,1}\cong C(T_{36,3}),&D_{36,2}\cong
C(T_{36,8}),&D_{36,3}\cong C(T_{36,5}),\\\ D_{36,4}\cong
C(T_{36,6}),&D_{36,5}\cong C(T_{36,7}),&D_{36,6}\cong C(T_{36,1}),\\\
D_{36,7}\cong C(T_{36,10}),&D_{36,11}\cong C(T_{36,4}),&D_{36,12}\cong
C(T_{36,2})\cong C(T_{36,9}).\end{array}$

For each code $Q_{36,i}$ ($i=1,2,\ldots,55$), we verified that there is a code
$F_{36,j}$ such that $Q_{36,i}\cong F_{36,j}$. In addition, we verified that
there is no code $F_{36,i}$ such that $F_{36,i}\cong C$ for each code
$C=C(T_{36,j})$ $(j=1,2,\ldots,8,10)$ and $C=D_{36,j}$ $(j=8,9,10)$. Hence, we
consider the following $272$ inequivalent near-extremal self-dual codes:

$\begin{array}[]{ll}C(T_{36,i})&(i=1,2,\ldots,8,10),\\\
D_{36,i}&(i=8,9,10),\\\ F_{36,i}&(i=1,2,\ldots,260),\end{array}$

in the next section.

##  4 Hadamard matrices in some near-extremal self-dual codes

In this section, we study the existence of Hadamard matrices formed by
codewords of weight $36$ in $C\in{\boldsymbol{C}_{36}}$, where

${{\boldsymbol{C}_{36}}}=\left\\{\begin{array}[]{l}C(T_{36,1}),C(T_{36,2}),\ldots,C(T_{36,8}),C(T_{36,10}),\\\
\qquad
D_{36,8},D_{36,9},D_{36,10},F_{36,1},F_{36,2},\ldots,F_{36,260}\end{array}\right\\}.$

###  4.1 Method

The following lemma is trivial.

######  Lemma 1.

Let $r_{1}$ and $r_{2}$ be vectors of $\\{1,-1\\}^{n}\subset\mathbb{Z}^{n}$.
Let $n_{1}(r_{i})$ denote the number of $1$’s in $r_{i}$ $(i=1,2)$. Suppose
that $n\equiv 0\pmod{4}$ and $r_{1}\neq\pm r_{2}$. If $r_{1}$ and $r_{2}$ are
orthogonal, then $n_{1}(r_{1})\equiv n_{1}(r_{2})\pmod{2}$.

We describe how to find all Hadamard matrices formed by codewords of weight
$36$ in a given near-extremal self-dual code $C$ of length $36$. Suppose that
$C$ has at least $72$ codewords of weight $36$. For the set $W(C)$ of
codewords of weight $36$ in $C$, we define the following sets:

$\displaystyle W_{1}(C)$ $\displaystyle=\\{x=(1,x_{2},x_{3},\ldots,x_{36})\mid
x\in W(C)\\},$ $\displaystyle W_{1,i}(C)$ $\displaystyle=\\{x\in W_{1}(C)\mid
n_{1}(x)\equiv i\\!\\!\\!\pmod{2}\\}\ (i=0,1).$

For a vector $x=(x_{1},x_{2},\ldots,x_{36})$ of $\mathbb{F}_{3}^{36}$, we
define the following:

$\overline{x}=(\overline{x_{1}},\overline{x_{2}},\ldots,\overline{x_{36}})\in\mathbb{Z}^{36},$
(5)

where $\overline{0}$, $\overline{1}$ and $\overline{2}$ are $0,1$ and
$-1\in\mathbb{Z}$, respectively. For a given set $W_{1,i}(C)$ $(i=0,1)$, a
simple undirected graph $\Gamma_{i}(C)$ is constructed as follows:

  1. (i)

the vertex set is $\\{\overline{x}\mid x\in
W_{1,i}(C)\\}\subset\mathbb{Z}^{36}$,

  2. (ii)

two vertices $x$ and $y$ are adjacent if $x$ and $y$ are orthogonal.

Clearly, a $36$-clique in $\Gamma_{i}(C)$ gives a Hadamard matrix for $i=0$
and $1$. By Lemma 1, all Hadamard matrices formed by codewords of $W_{1}(C)$
are obtained by finding all $36$-cliques in $\Gamma_{i}(C)$ for $i=0$ and $1$.
For any Hadamard matrix $H$ formed by codewords of weight $36$ in $C$, it
follows that there is a Hadamard matrix $H^{\prime}$ constructed by the above
method satisfying that $H\cong H^{\prime}$.

###  4.2 Results

By the above method, for each code $C\in{\boldsymbol{C}_{36}}$, we found all
distinct Hadamard matrices formed by codewords of weight $36$ in $C$. The
results are given by using the following notations:

$\displaystyle N_{i}(C)$ $\displaystyle=\text{ the number of $36$-cliques in
$\Gamma_{i}(C)$ $(i=0,1)$},$ $\displaystyle N(C)$ $\displaystyle=\text{ the
number of inequivalent Hadamard matrices in $C$},$
$\displaystyle\mathcal{T}(C)$
$\displaystyle=(|W_{1,0}(C)|,|W_{1,1}(C)|,N_{0}(C),N_{1}(C),N(C)).$

Firstly, for each code $C\in{\boldsymbol{C}_{36,1}}$, the tuples
$\mathcal{T}(C)$ are listed in Table 5, where

${\boldsymbol{C}_{36,1}}=\\{C(T_{36,1}),C(T_{36,2}),\ldots,C(T_{36,8}),C(T_{36,10}),\\\
D_{36,8},D_{36,9},D_{36,10}\\}.$

Since $N(C(T_{36,i}))=1$ $(i=1,3,4,\ldots,8,10)$, we denote by $H_{T,i}$ the
Hadamard matrix formed by codewords of weight $36$ in $C(T_{36,i})$. It is
easy to see that $T_{36,i}=H_{T,i}$ for $i=1,3,4,\ldots,8,10$. Since
$N(C(T_{36,2}))=2$ and $C(T_{36,2})\cong C(T_{36,9})$, $T_{36,2}$ is one of
the two inequivalent Hadamard matrices formed by codewords of weight $36$ in
$C(T_{36,2})$, and the other Hadamard matrix $H$ satisfies $H\cong T_{36,9}$.
Therefore, we have the following:

######  Proposition 2.

  1. (i)

There are ten inequivalent Hadamard matrices formed by codewords of weight
$36$ in the nine codes $C(T_{36,i})$ $(i=1,2,\ldots,8,10)$.

  2. (ii)

There is no Hadamard matrix formed by codewords of weight $36$ in the three
codes $D_{36,i}$ $(i=8,9,10)$.

Table 5: Tuples $\mathcal{T}(C)$ for $C\in{\boldsymbol{C}_{36,1}}$
---
$C$ | $\mathcal{T}(C)$
$C(T_{36,i})$ $(i=1,3,10)$ | $(308,68,1,0,1)$
$C(T_{36,2})$ | $(104,68,2,0,2)$
$C(T_{36,4})$ | $(104,68,1,0,1)$
$C(T_{36,i})$ $(i=5,6,7,8)$ | $(104,272,1,0,1)$
$D_{36,i}$ $(i=8,9,10)$ | $(204,36,0,0,0)$

Secondly, for each code
$C\in{\boldsymbol{C}_{36}}\setminus{\boldsymbol{C}_{36,1}}$, we found all
distinct Hadamard matrices formed by codewords of weight $36$ in $C$ by the
above method. It holds that

$|\\{\mathcal{T}(F_{36,1}),\mathcal{T}(F_{36,2}),\ldots,\mathcal{T}(F_{36,260})\\}|=153,$

where the $153$ distinct tuples $\mathcal{T}(F_{36,i})$ are listed in Tables 9
and 10. For each tuple $\mathcal{T}$ in the tables, we list all values $i$
such that $\mathcal{T}=\mathcal{T}(F_{36,i})$. For each $i=1,2,\ldots,260$, we
denote by $H_{F,i,i_{j}}$ the Hadamard matrices formed by codewords of weight
$36$ in the $260$ codes $F_{36,i}$, where $i_{j}=1,2,\ldots,N(F_{36,i})$. We
verified that

$\begin{array}[]{ll}H_{F,11,1}\cong H_{F,67,1}\cong
H_{F,139,1},&H_{F,27,1}\cong H_{F,83,1}\cong H_{F,237,1},\\\ H_{F,38,2}\cong
H_{F,79,1}\cong H_{F,177,1},&H_{F,49,2}\cong H_{F,168,1}\cong H_{F,169,1},\\\
H_{F,56,2}\cong H_{F,115,1}\cong H_{F,213,1},&H_{F,64,1}\cong H_{F,172,1}\cong
H_{F,174,1},\end{array}$

and there is no other pair of equivalent Hadamard matrices among the Hadamard
matrices $H_{F,i,i_{j}}$. Therefore, we have the following:

######  Proposition 3.

There are $89$ inequivalent Hadamard matrices formed by codewords of weight
$36$ in the $260$ codes $F_{36,i}$ $(i=1,2,\ldots,260)$.

######  Remark 4.

The above $89$ inequivalent Hadamard matrices can be obtained electronically
from https://www.math.is.tohoku.ac.jp/~mharada/F3-36/Hmat-F.txt.

We consider equivalences among the Hadamard matrices formed by codewords of
weight $36$ in the $272$ codes of ${\boldsymbol{C}_{36}}$. The orders of the
automorphism groups $\operatorname{Aut}(H_{F,i,i_{j}})$ of $H_{F,i,i_{j}}$ are
listed in Table 6. From the table, there is no pair of inequivalent Hadamard
matrices among the $99$ inequivalent Hadamard matrices given in Propositions 2
and 3.

Table 6: Orders of the automorphism groups $\operatorname{Aut}(H_{F,i,i_{j}})$
---
$|\operatorname{Aut}(H_{F,i,i_{j}})|$ | $(i,i_{j})$
8 | $(8,1),(18,2),(23,1),(23,2),(168,2),(168,3)$
12 | $(7,1),(39,1)$
16 | $(37,1),(37,3)$
24 | $(8,2),(18,3)$
36 | $(4,1),(4,2),(9,1),(11,1),(12,1),(12,2),(13,1),(18,1),(26,1),$
| $(26,2),(27,1),(28,1),(30,1),(35,1),(38,1),(38,2),(42,1),$
| $(53,1),(56,1),(56,2),(60,1),(61,1),(66,1),(73,1),(96,1),$
| $(103,1),(104,1),(110,1),(111,1),(112,1),(117,1),(120,1),$
| $(130,1),(133,1),(148,1),(149,1),(166,1),(173,1),(176,1),$
| $(176,2),(178,1),(193,1),(197,1),(201,1),(208,1),(210,1),$
| $(213,2),(226,1),(227,1),(230,1),(234,1),(235,1),(245,1),$
| $(246,1),(247,1),(254,1)$
72 | $(23,3),(49,1),(49,2),(54,1),(54,2),(64,1),(68,1),(90,1),$
| $(108,1),(108,2),(143,1),(143,2),(164,1),(168,4),(184,1),$
| $(188,1),(215,1),(224,1),(260,1)$
144 | $(37,2),(194,1)$

Now we consider equivalences among the $89$ inequivalent Hadamard matrices
$H_{F,i,j}$ and their transposes. We verified that $H_{F,s,t}^{T}$ is
equivalent to none of the $89$ inequivalent Hadamard matrices $H_{F,i,j}$ for
each $(s,t)\in\mathcal{S}_{s,t}$ and there is a Hadamard matrix $H_{F,i,j}$
such that $H_{F,i,j}\cong H_{F,s,t}^{T}$ for each
$(s,t)\not\in\mathcal{S}_{s,t}$, where

$\mathcal{S}_{s,t}=\left\\{\begin{array}[]{l}(7,1),(8,1),(8,2),(18,2),(18,3),(23,1),(23,2),\\\
(37,1),(37,3),(39,1),(168,2),(168,3),(260,1)\end{array}\right\\}.$ (6)

By the Magma function HadamardDatabase [2], one can construct $219$
inequivalent Hadamard matrices of order $36$, including the ten Hadamard
matrices in [16]. We verified that there is no pair of inequivalent Hadamard
matrices among the $89$ inequivalent Hadamard matrices $H_{F,i,j}$ given in
Proposition 3, the $13$ inequivalent Hadamard matrices $H_{F,s,t}^{T}$ for
$(s,t)\in\mathcal{S}_{s,t}$ in (6) and the above $219$ inequivalent Hadamard
matrices in [2].

###  4.3 Related codes

For the $89$ inequivalent Hadamard matrices $H_{F,i,j}$ given in Proposition
3, we calculated the dimensions $\dim(C(H_{F,i,j}))$ and the minimum weights
$d(C(H_{F,i,j}))$ of their codes $C(H_{F,i,j})$. The results are as follows:

$\displaystyle\dim(C(H_{F,i,j}))$ $\displaystyle=\begin{cases}16&\text{ if
}(i,j)=(11,1),(27,1),(37,3),(38,2),\\\
&\qquad(39,1),(49,2),(56,2),(64,1),(168,2),\\\ 18&\text{
otherwise,}\end{cases}$ $\displaystyle d(C(H_{F,i,j}))$
$\displaystyle=\begin{cases}12&\text{ if }(i,j)=(11,1),(38,2),\\\ 9&\text{
otherwise.}\end{cases}$

In addition, we verified that

$d(C(H_{F,s,t}^{T}))=\begin{cases}12&\text{ if }(s,t)=(260,1),\\\ 9&\text{
otherwise,}\end{cases}$

along with $C(H_{F,260,1}^{T})\cong P_{36}$. Since
$|\operatorname{Aut}(H_{F,260,1})|=72$ (see Table 6), we have the following:

######  Proposition 5.

Let $H^{\prime}$ be the second Hadamard matrix formed by codewords of weight
$36$ in the Pless symmetry code $P_{36}$, which is given in [17, Theorem 2.4
(i)]. Then the Hadamard matrix $H_{F,260,1}^{T}$ is equivalent to
$H^{\prime}$.

The above proposition indicates a relationship between the two codes $P_{36}$
and $C(H_{F,260,1})$ via the Hadamard matrix $H^{\prime}$ given in [17,
Theorem 2.4 (i)].

We end this note by giving some observation on the code $C(H_{F,260,1})$. Let
$W(C(H_{F,260,1}))$ be the set of $144$ codewords of weight $36$ in
$C(H_{F,260,1})$. Since

$\mathcal{T}(C(H_{F,260,1}))=(36,36,1,1,1)$

(see Table 10), it follows that there are $h_{i}$ and $h^{\prime}_{i}\
(i=1,\dots,36)$ such that

$W(C(H_{F,260,1}))=\\{h_{i},2h_{i},h^{\prime}_{i},2h^{\prime}_{i}\mid
i=1,2,\ldots,36\\}$

and the following two $36\times 36$ matrices

$\left(\begin{array}[]{ccc}&\overline{h_{1}}&\\\ &\vdots&\\\
&\overline{h_{36}}&\\\ \end{array}\right)\text{ and
}\left(\begin{array}[]{ccc}&\overline{h^{\prime}_{1}}&\\\ &\vdots&\\\
&\overline{h^{\prime}_{36}}&\\\ \end{array}\right)$

are Hadamard matrices (see (5) for the notation $\overline{h_{i}}$). This
means that the $144$ codewords of weight $36$ in $C(H_{F,260,1})$ correspond
to the rows and their negatives of two Hadamard matrices. It is worthwhile to
determine whether there is a near-extremal self-dual code with $A_{9}=816$
satisfying the condition that the $72$ codewords of weight $36$ correspond to
the rows and their negatives of a Hadamard matrix.

Acknowledgments. This work was supported by JSPS KAKENHI Grant Number
19H01802.

## References

  * [1] M. Araya and M. Harada, Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes, Des. Codes Cryptogr., (to appear), https://doi.org/10.1007/s10623-022-01172-7.
  * [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265.
  * [3] J.H. Conway, V. Pless and N.J.A. Sloane, Self-dual codes over $GF(3)$ and $GF(4)$ of length not exceeding $16$, IEEE Trans. Inform. Theory 25 (1979), 312–322.
  * [4] M. Harada, W. Holzmann, H. Kharaghani and M. Khorvash, Extremal ternary self-dual codes constructed from negacirculant matrices, Graphs Combin. 23 (2007), 401–417.
  * [5] M. Harada and A. Munemasa, A complete classification of ternary self-dual codes of length $24$, J. Combin. Theory Ser. A 116 (2009), 1063–1072.
  * [6] W.C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl. 11 (2005), 451–490.
  * [7] H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order $32$, J. Combin. Designs 21 (2013), 212–221.
  * [8] H. Kimura, Classification of Hadamard matrices of order $28$, Discrete Math. 133 (1994), 171–180.
  * [9] C.L. Mallows, V. Pless and N.J.A. Sloane, Self-dual codes over $GF(3)$, SIAM J. Appl. Math. 31 (1976), 649–666.
  * [10] C.L. Mallows and N.J.A. Sloane, An upper bound for self-dual codes, Inform. Control 22 (1973), 188–200.
  * [11] V. Pless, Symmetry codes over $GF(3)$ and new five-designs, J. Combin. Theory Ser. A 12 (1972), 119–142.
  * [12] V. Pless, N.J.A. Sloane and H.N. Ward, Ternary codes of minimum weight $6$ and the classification of the self-dual codes of length $20$, IEEE Trans. Inform. Theory 26 (1980), 305–316.
  * [13] E. Rains and N.J.A. Sloane, “Self-dual codes,” Handbook of Coding Theory, V.S. Pless and W.C. Huffman (Editors), Elsevier, Amsterdam, pp. 177–294, 1998.
  * [14] J. Seberry, Library of Hadamard Matrices, https://documents.uow.edu.au/~jennie/hadamard.html.
  * [15] J. Seberry and M. Yamada, Hadamard Matrices: Constructions using Number Theory and Linear Algebra, Wiley, NJ, 2020.
  * [16] V.D. Tonchev, Hadamard matrices of order $36$ with automorphisms of order $17$, Nagoya Math. J. 104 (1986), 163–174.
  * [17] V.D. Tonchev, On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs, Des. Codes Cryptogr. 90 (2022), 2753–2762.

Table 7: Four-negacirculant codes $F_{36,i}$ $(i=1,2,\ldots,132)$
---
$i$ | $r_{A}$ | $r_{B}$ | $i$ | $r_{A}$ | $r_{B}$ | $i$ | $r_{A}$ | $r_{B}$ | $i$ | $r_{A}$ | $r_{B}$
1 | 100211222 | 012000012 | 34 | 120121102 | 111000001 | 67 | 112112202 | 201210201 | 100 | 110002201 | 121022020
2 | 100121012 | 221100001 | 35 | 101120120 | 112200010 | 68 | 122000102 | 122111221 | 101 | 100220221 | 020221020
3 | 110200102 | 221100021 | 36 | 122002022 | 022200012 | 69 | 112012122 | 102200000 | 102 | 101102222 | 202210221
4 | 120201102 | 111000011 | 37 | 120012020 | 111000000 | 70 | 120220222 | 001200022 | 103 | 102000222 | 211022002
5 | 121202010 | 222200100 | 38 | 110020121 | 202100021 | 71 | 112212022 | 122012010 | 104 | 100210012 | 111200011
6 | 111201011 | 201200100 | 39 | 111100210 | 122200010 | 72 | 110202212 | 201211201 | 105 | 100212122 | 210100020
7 | 112010202 | 012200021 | 40 | 122002002 | 122200021 | 73 | 112121021 | 200222210 | 106 | 100110210 | 212220010
8 | 102101012 | 111100002 | 41 | 121002022 | 220100120 | 74 | 121011111 | 021110110 | 107 | 101210102 | 212221011
9 | 122202021 | 020200011 | 42 | 102222212 | 121100110 | 75 | 121220221 | 010011121 | 108 | 100002222 | 222201002
10 | 121111212 | 122200010 | 43 | 120122122 | 212200102 | 76 | 111122222 | 121220111 | 109 | 110012102 | 121221120
11 | 102102221 | 021200010 | 44 | 121201221 | 111200110 | 77 | 111221001 | 101221210 | 110 | 112010221 | 011001010
12 | 112211120 | 001200010 | 45 | 121021012 | 222100121 | 78 | 101012100 | 121112000 | 111 | 102211212 | 201110210
13 | 122012122 | 222100022 | 46 | 100122122 | 101100001 | 79 | 122120022 | 012022121 | 112 | 110102222 | 211011120
14 | 121022021 | 021100001 | 47 | 121011120 | 110200020 | 80 | 110001101 | 011222020 | 113 | 102021202 | 201002102
15 | 110010221 | 021200021 | 48 | 111020210 | 122100010 | 81 | 120110222 | 221110110 | 114 | 112101122 | 220212020
16 | 101001011 | 121200011 | 49 | 112202020 | 121100010 | 82 | 120021122 | 120220121 | 115 | 122012221 | 120021101
17 | 121012112 | 122200120 | 50 | 121020201 | 212200001 | 83 | 112210212 | 122102100 | 116 | 111120212 | 001222201
18 | 121122211 | 222100100 | 51 | 102201002 | 221200011 | 84 | 122120211 | 101122100 | 117 | 100001211 | 110120210
19 | 110211012 | 212200111 | 52 | 122012221 | 121200110 | 85 | 101001221 | 012110020 | 118 | 110122002 | 200120210
20 | 122102212 | 121100101 | 53 | 120212112 | 112200101 | 86 | 111212210 | 112020110 | 119 | 122212122 | 010202202
21 | 122222022 | 122100101 | 54 | 122020202 | 220200110 | 87 | 102210120 | 201200210 | 120 | 112100001 | 111012200
22 | 120220112 | 121100121 | 55 | 101120122 | 101200100 | 88 | 122002201 | 100101120 | 121 | 102211111 | 200010100
23 | 100211101 | 212200001 | 56 | 122020211 | 121200111 | 89 | 121122212 | 021202001 | 122 | 122201011 | 101200010
24 | 120021202 | 212200020 | 57 | 121010122 | 111200121 | 90 | 101112222 | 101101202 | 123 | 121111211 | 221121110
25 | 102201101 | 112100020 | 58 | 121020212 | 221100112 | 91 | 122100221 | 100222000 | 124 | 122021021 | 002002210
26 | 122001012 | 212200001 | 59 | 122011022 | 121100112 | 92 | 110101202 | 210022100 | 125 | 122222002 | 121110210
27 | 110222022 | 102200020 | 60 | 110121021 | 122100111 | 93 | 120220211 | 200010101 | 126 | 100110202 | 021021120
28 | 121210101 | 101200010 | 61 | 102121202 | 001200011 | 94 | 112111122 | 000212021 | 127 | 110221220 | 020020210
29 | 102001112 | 110200012 | 62 | 100102220 | 110000010 | 95 | 110102001 | 012221001 | 128 | 122011121 | 202121100
30 | 111022122 | 212100022 | 63 | 100202222 | 221222120 | 96 | 102020022 | 112021001 | 129 | 111001001 | 021222100
31 | 120120021 | 222100020 | 64 | 121201202 | 012121210 | 97 | 122222002 | 100010101 | 130 | 100201202 | 120020111
32 | 112220201 | 202100010 | 65 | 110112122 | 010120221 | 98 | 120101210 | 110001110 | 131 | 102110221 | 020002201
33 | 120210012 | 222100001 | 66 | 111201022 | 100122211 | 99 | 121122010 | 121112100 | 132 | 102010022 | 120010211
Table 8: Four-negacirculant codes $F_{36,i}$ $(i=133,134,\ldots,260)$
---
$i$ | $r_{A}$ | $r_{B}$ | $i$ | $r_{A}$ | $r_{B}$ | $i$ | $r_{A}$ | $r_{B}$ | $i$ | $r_{A}$ | $r_{B}$
133 | 111022202 | 211011021 | 165 | 112210121 | 002000101 | 197 | 112122011 | 212200011 | 229 | 102011021 | 211202111
134 | 102111101 | 101000101 | 166 | 100221122 | 111002121 | 198 | 102010111 | 221200010 | 230 | 100000011 | 120110200
135 | 120022101 | 201022200 | 167 | 120121011 | 221212020 | 199 | 111000202 | 101011110 | 231 | 121212012 | 001112120
136 | 121120221 | 001002010 | 168 | 100002222 | 020121102 | 200 | 120012221 | 021111220 | 232 | 101212002 | 122210211
137 | 112010112 | 111000020 | 169 | 101011212 | 022120111 | 201 | 112011201 | 122022210 | 233 | 100212202 | 011110020
138 | 121121222 | 002022110 | 170 | 100020221 | 221200011 | 202 | 102011222 | 102220122 | 234 | 100211021 | 112102211
139 | 121020101 | 011202010 | 171 | 120010222 | 211202221 | 203 | 100000202 | 022120100 | 235 | 112022101 | 211111001
140 | 101002122 | 222201000 | 172 | 102121100 | 001200000 | 204 | 111202101 | 221021101 | 236 | 100200021 | 002022200
141 | 121220021 | 221102110 | 173 | 122110202 | 212222001 | 205 | 112111022 | 010010200 | 237 | 110012021 | 011200210
142 | 122020021 | 221122120 | 174 | 121122121 | 002211100 | 206 | 110022112 | 200020110 | 238 | 100122002 | 112200021
143 | 122020112 | 202001200 | 175 | 120122212 | 102101110 | 207 | 100210111 | 111011000 | 239 | 120001212 | 022220001
144 | 100100202 | 101012000 | 176 | 120200202 | 010200010 | 208 | 100110012 | 220022202 | 240 | 102111011 | 012112120
145 | 101222001 | 011202010 | 177 | 102112100 | 221110000 | 209 | 112110002 | 002012210 | 241 | 101022220 | 210212000
146 | 100000212 | 001200120 | 178 | 112110010 | 102110100 | 210 | 120110022 | 210200120 | 242 | 112001122 | 012110111
147 | 122022012 | 121110120 | 179 | 101210201 | 022112000 | 211 | 100201111 | 011011200 | 243 | 101211112 | 001221120
148 | 110100222 | 111220000 | 180 | 101011112 | 112202202 | 212 | 112120021 | 010020110 | 244 | 102102002 | 112200120
149 | 120221122 | 101120210 | 181 | 122212022 | 002121201 | 213 | 101100022 | 020200200 | 245 | 121202121 | 010111201
150 | 112002101 | 220102200 | 182 | 112010012 | 012221111 | 214 | 101100111 | 221100010 | 246 | 102111211 | 002200010
151 | 121210002 | 220021020 | 183 | 120212002 | 112011121 | 215 | 102001021 | 121200210 | 247 | 102112212 | 201200121
152 | 122222222 | 122102212 | 184 | 101010001 | 021012000 | 216 | 112201212 | 102210220 | 248 | 100221202 | 100200121
153 | 121021011 | 220122220 | 185 | 102220202 | 002011021 | 217 | 120020221 | 010111020 | 249 | 121122222 | 111201111
154 | 100001012 | 002200210 | 186 | 120211010 | 120111000 | 218 | 100200222 | 002212201 | 250 | 122221122 | 111100020
155 | 121112011 | 022000001 | 187 | 110202212 | 012121220 | 219 | 110210021 | 200202011 | 251 | 102021221 | 012112101
156 | 100212102 | 200022120 | 188 | 111021022 | 122002000 | 220 | 122110102 | 002002101 | 252 | 120211202 | 021200010
157 | 112201022 | 200001201 | 189 | 120011011 | 121000120 | 221 | 111012120 | 212002000 | 253 | 111100010 | 021100000
158 | 111100012 | 022012010 | 190 | 110220021 | 022200110 | 222 | 100210202 | 101002111 | 254 | 112200020 | 111000000
159 | 102102020 | 010011000 | 191 | 122100002 | 112201200 | 223 | 120200122 | 021210100 | 255 | 122200010 | 222000000
160 | 112121022 | 001110111 | 192 | 112210012 | 100120002 | 224 | 120121120 | 102221110 | 256 | 101010120 | 110000010
161 | 100202122 | 222011121 | 193 | 111201102 | 110200001 | 225 | 100021112 | 101000211 | 257 | 101112122 | 212221000
162 | 110221112 | 200211021 | 194 | 122001120 | 010100000 | 226 | 110020002 | 000101011 | 258 | 120012001 | 001212110
163 | 101001202 | 120111001 | 195 | 101211222 | 120220201 | 227 | 110110221 | 111100221 | 259 | 112210201 | 220200020
164 | 102210011 | 002001000 | 196 | 102112001 | 121200001 | 228 | 102120212 | 212121200 | 260 | 112101021 | 200000000
Table 9: Tuples $\mathcal{T}(F_{36,i})$
---
$i$ | $\mathcal{T}(F_{36,i})$ | $i$ | $\mathcal{T}(F_{36,i})$ | $i$ | $\mathcal{T}(F_{36,i})$
$1,6$ | $(144,264,0,0,0)$ | $35$ | $(120,216,0,1,1)$ | $80,81,87,98,99$ | $(264,36,0,0,0)$
$2,5$ | $(408,0,0,0,0)$ | $37$ | $(84,252,0,19,3)$ | $83$ | $(216,84,1,0,1)$
$3$ | $(372,36,0,0,0)$ | $38$ | $(72,264,1,1,2)$ | $88,94$ | $(48,252,0,0,0)$
$4$ | $(264,144,1,1,2)$ | $39$ | $(288,48,3,0,1)$ | $89$ | $(36,264,0,0,0)$
$7$ | $(252,144,3,0,1)$ | $40,44$ | $(108,228,0,0,0)$ | $91$ | $(156,144,0,0,0)$
$8$ | $(288,108,12,0,2)$ | $41$ | $(144,192,0,0,0)$ | $95,101$ | $(84,216,0,0,0)$
$9$ | $(360,12,1,0,1)$ | $42$ | $(180,156,1,0,1)$ | $103$ | $(84,216,0,1,1)$
$10$ | $(144,228,0,0,0)$ | $43$ | $(264,72,0,0,0)$ | $105,109,113$ | $(108,180,0,0,0)$
$11$ | $(288,84,1,0,1)$ | $45$ | $(72,264,0,0,0)$ | $106,107,116$ | $(216,72,0,0,0)$
$12$ | $(264,108,4,0,2)$ | $46$ | $(144,180,0,0,0)$ | $108$ | $(216,72,2,0,2)$
$13$ | $(84,288,0,1,1)$ | $47$ | $(252,72,0,0,0)$ | $110$ | $(216,72,1,0,1)$
$14$ | $(300,72,0,0,0)$ | $48,52$ | $(288,36,0,0,0)$ | $111$ | $(180,108,1,0,1)$
$15$ | $(336,36,0,0,0)$ | $49$ | $(144,180,0,2,2)$ | $112,117,120$ | $(36,252,0,1,1)$
$16$ | $(72,300,0,0,0)$ | $50,57$ | $(180,144,0,0,0)$ | $114$ | $(180,108,0,0,0)$
$17$ | $(216,156,0,0,0)$ | $51,55,58,59$ | $(216,108,0,0,0)$ | $115$ | $(108,180,0,1,1)$
$18$ | $(84,288,0,14,3)$ | $53$ | $(36,288,0,1,1)$ | $118,119$ | $(144,144,0,0,0)$
$19,21$ | $(36,336,0,0,0)$ | $54$ | $(36,288,0,2,2)$ | $121$ | $(72,216,0,0,0)$
$20$ | $(228,144,0,0,0)$ | $56$ | $(252,72,2,0,2)$ | $122$ | $(192,72,0,0,0)$
$22$ | $(192,180,0,0,0)$ | $60$ | $(288,36,1,0,1)$ | $123,129,132,137,141,154$ | $(144,120,0,0,0)$
$23$ | $(252,108,19,0,3)$ | $61,66$ | $(264,36,1,0,1)$ | $124,138$ | $(36,228,0,0,0)$
$24$ | $(216,144,0,0,0)$ | $62,69,71,75,77$ | $(144,156,0,0,0)$ | $125,128,144,146$ | $(84,180,0,0,0)$
$25$ | $(72,288,0,0,0)$ | $63,65,72,84,85,97,100$ | $(192,108,0,0,0)$ | $126,140,152$ | $(72,192,0,0,0)$
$26$ | $(36,324,0,4,2)$ | $64,90$ | $(144,156,0,1,1)$ | $127,134,150$ | $(108,156,0,0,0)$
$27$ | $(156,180,0,1,1)$ | $67,96$ | $(156,144,0,1,1)$ | $130,139$ | $(84,180,0,1,1)$
$28$ | $(300,36,2,0,1)$ | $68$ | $(72,228,0,1,1)$ | $131,135$ | $(120,144,0,0,0)$
$29$ | $(120,216,0,0,0)$ | $70,74,92,93,102$ | $(108,192,0,0,0)$ | $133$ | $(120,144,0,1,1)$
$30$ | $(288,48,1,0,1)$ | $73$ | $(48,252,0,1,1)$ | $136,147,153$ | $(180,84,0,0,0)$
$31,32$ | $(156,180,0,0,0)$ | $76,82,86$ | $(180,120,0,0,0)$ | $142,151$ | $(228,36,0,0,0)$
$33,36$ | $(216,120,0,0,0)$ | $78$ | $(216,84,0,0,0)$ | $143$ | $(72,192,0,2,2)$
$34$ | $(48,288,0,0,0)$ | $79,104$ | $(192,108,1,0,1)$ | $145$ | $(156,108,0,0,0)$
Table 10: Tuples $\mathcal{T}(F_{36,i})$
---
$i$ | $\mathcal{T}(F_{36,i})$ | $i$ | $\mathcal{T}(F_{36,i})$ | $i$ | $\mathcal{T}(F_{36,i})$
$148$ | $(228,36,1,0,1)$ | $191$ | $(48,180,0,0,0)$ | $230$ | $(144,48,1,0,1)$
$149$ | $(192,72,1,0,1)$ | $193$ | $(192,36,1,0,1)$ | $231$ | $(0,192,0,0,0)$
$155,157,167$ | $(144,108,0,0,0)$ | $194$ | $(144,84,1,0,1)$ | $234$ | $(84,108,0,1,1)$
$156,158,160,161,165,171$ | $(108,144,0,0,0)$ | $197$ | $(36,180,0,1,1)$ | $235,237$ | $(108,84,1,0,1)$
$159,163$ | $(216,36,0,0,0)$ | $198,199,204,212$ | $(108,108,0,0,0)$ | $238,239,241$ | $(144,36,0,0,0)$
$162,170$ | $(72,180,0,0,0)$ | $200$ | $(36,180,0,0,0)$ | $240$ | $(36,144,0,0,0)$
$164$ | $(72,180,0,1,1)$ | $201$ | $(108,108,0,1,1)$ | $242,244$ | $(108,72,0,0,0)$
$166$ | $(216,36,1,0,1)$ | $202,205,209$ | $(180,36,0,0,0)$ | $243$ | $(72,108,0,0,0)$
$168$ | $(252,0,20,0,4)$ | $203,206,211$ | $(144,72,0,0,0)$ | $245$ | $(144,36,2,0,1)$
$169$ | $(252,0,1,0,1)$ | $207$ | $(72,144,0,0,0)$ | $246,247$ | $(120,36,1,0,1)$
$172,188$ | $(0,228,0,1,1)$ | $208$ | $(108,108,1,0,1)$ | $248$ | $(108,48,0,0,0)$
$173$ | $(72,156,0,1,1)$ | $210$ | $(180,36,1,0,1)$ | $249$ | $(72,84,0,0,0)$
$174$ | $(228,0,1,0,1)$ | $213$ | $(144,72,1,1,2)$ | $250,251$ | $(72,72,0,0,0)$
$175,180,187,189$ | $(156,72,0,0,0)$ | $214,218,219,225,236$ | $(108,84,0,0,0)$ | $252$ | $(72,48,0,0,0)$
$176$ | $(36,192,0,2,2)$ | $215,224,226,227$ | $(36,156,0,1,1)$ | $253$ | $(48,72,0,0,0)$
$177$ | $(156,72,1,0,1)$ | $216$ | $(120,72,0,0,0)$ | $254$ | $(12,108,0,2,1)$
$178$ | $(48,180,0,1,1)$ | $217,233$ | $(84,108,0,0,0)$ | $255$ | $(120,0,0,0,0)$
$179,182,185,192,195$ | $(108,120,0,0,0)$ | $220,222,228,232$ | $(36,156,0,0,0)$ | $256$ | $(72,12,0,0,0)$
$181,183,196$ | $(120,108,0,0,0)$ | $221$ | $(144,48,0,0,0)$ | $257,258$ | $(36,48,0,0,0)$
$184$ | $(108,120,0,1,1)$ | $223$ | $(48,144,0,0,0)$ | $259$ | $(72,0,0,0,0)$
$186,190$ | $(72,156,0,0,0)$ | $229$ | $(72,120,0,0,0)$ | $260$ | $(36,36,1,1,1)$