# Combinatorial Constructions of Optimal Quaternary Additive Codes

Chaofeng Guan, Jingjie Lv, Gaojun Luo, Zhi Ma Chaofeng Guan and Zhi Ma are
with the Henan Key Laboratory of Network Cryptography Technology, Zhengzhou
450001, China. (e-mails<EMAIL_ADDRESS>LWtglx2023@outlook.com).Jingjie Lv
is with Tsinghua Shenzhen International Graduate School, Tsinghua University,
Shenzhen 518055, China (email: juxianljj@sz.tsinghua.edu.cn). Gaojun Luo is
with the Department of Mathematics, Nanjing University of Aeronautics and
Astronautics, Nanjing 210016, China (e-mail: gjluo1990@163.com).Chaofeng Guan
is supported by the National Natural Science Foundation of China under Grant
U21A20428, and the National Key Research and Development Program of China
under Grant 2021YFB3100100. Zhi Ma is supported by the National Natural
Science Foundation of China under Grant 61972413. Gaojun Luo is supported by
the Natural Science Foundation of Jiangsu Province, under Grant BK20230867 and
National Natural Science Foundation of China under Grant 12171241.

###### Abstract

This paper aims to construct optimal quaternary additive codes with non-
integer dimensions. Firstly, we propose combinatorial constructions of
quaternary additive constant-weight codes, alongside additive anticode
construction. Subsequently, we propose generalized Construction X, which
facilitates the construction of non-integer dimensional optimal additive codes
from linear codes. Then, we construct ten classes of optimal quaternary non-
integer dimensional additive codes through these two methods. As an
application, we also determine the optimal additive $[n,3.5,n-t]_{4}$ codes
for all $t$ with variable $n$, except for $t=6,7,12$.

###### Index Terms:

Quaternary additive code, additive constant-weight code, additive anticode
construction, generalized Construction X.

††publicationid: pubid:

##  I Introduction

Additive codes are closed under vector addition but not necessarily under
scalar multiplication. All linear codes are additive, but additive codes are
not necessarily linear. Therefore, the parameters of additive codes may be
better than optimal linear codes. Moreover, additive codes are an interesting
class of error-correcting codes and have significant applications in quantum
information [1, 2], computer memory systems [3, 4, 5], deep space
communication [6], protection against side-channel attacks [7, 8], and
distance-regular graph [9].

In [10], Blokhuis and Brouwer determined the parameters of optimal quaternary
additive codes of lengths up to 12. Later in [11, 12, 13, 14, 15], much work
has been performed on quaternary additive codes with small lengths, resulting
in a general determination of the parameters of optimal additive codes of
lengths up to 15. In [16], Bierbrauer et al. determined the parameters of all
2.5-dimensional optimal quaternary additive codes by a projective geometry
approach. However, as the dimensions increase, the number of involved optimal
additive codes increases exponentially. For instance, for $k=2.5$, the
parameters of optimal quaternary 2.5-dimensional additive codes can be
determined by only three non-trivial key codes, while the problem is much more
complicated for $k\geq 3.5$. In [17, 18, 19], many optimal quaternary
3.5-dimensional additive codes were obtained by computer-supported methods.
However, these work could only partially determine the optimal quaternary
3.5-dimensional additive codes of lengths from 28 to 254. A solution to the
problem of constructing optimal quaternary 3.5-dimensional or higher-
dimensional additive codes requires more efficient methods and theories.

The Griesmer bound is essential in determining the performance of linear
codes, and the following is the generalization of the Griesmer bound for
additive codes.

###### Lemma 1

(Additive Griesmer Bound, [18]) If $C_{a}$ is a quaternary additive
$[n,k,d]_{4}$ code with $k\geq 1$, then

$3n\geq
g(2k,2d)=\sum\limits_{i=0}^{2k-1}\left\lceil\frac{d}{2^{i-1}}\right\rceil.$
(1)

To distinguish optimal scenarios concerning the above bound, we define
$C_{n,t}$ as additive $[n,k,n-t]_{4}$ code, where $n$ and $t$ are positive
integers. By changing the values of $n$ and $t$, $C_{n,t}$ can represent all
additive codes with dimension $k$. As in the linear case, an additive code
satisfies the additive Griesmer bound with equality is called additive
Griesmer code. For a fixed $n$, if $C_{n,t}$ satisfies

$3n<g(2k,2(n-t)+2),$ (2)

then we call $C_{n,t}$ an additive Griesmer distance-optimal (GDO) code. If no
such code $C_{n,t}$ exists, then we refer to the code with maximum $t$ that
exists as an additive distance-optimal (DO) code. For a fixed $t$, if
$C_{n,t}$ satisfies

$3n+3<g(2k,2(n-t)+2),$ (3)

i.e., there is no $[n+1,k,n-t+1]_{4}$ code, then we call $C_{n,t}$ as an
additive Griesmer puncture-optimal (GPO) code; if such a code $C_{n,t}$ does
not exist, then we call the longest GDO (or DO) code an additive puncture-
optimal (PO) code. This is, all additive GDO (or DO) codes can be punctured
from additive GPO (or PO) codes; otherwise, they are also GPO (or PO).
Therefore, for a specific dimension $k$, when we want to construct optimal
additive code $C_{n,t}$ for all $t$, we only need to construct all
$k$-dimensional GPO and PO codes. These facts suggest that GPO and PO codes
are the most critical optimal codes.

Few-weight codes are an interesting class of error-correcting codes. They have
significant applications in strongly regular graphs [20, 21], communication
[22], cryptography [23, 23], quantum information [24], and can be decoded
using the minimum distance decoding method [22]. Recently, researchers have
done much work on the construction and application of few-weight codes in [25,
26, 27, 28, 29, 30, 31, 32]. It is worth noting that our GPO quaternary
additive codes are all few-weight codes; see Table I.

The main objective of this paper is to construct optimal quaternary additive
codes with non-integer dimensions using combinatorial methods. As a result,
many classes of optimal quaternary additive codes are obtained, and their
weight distributions are also determined. The contributions of this paper can
be summarized as follows.

  1. 1.

We present three methods for constructing a class of quaternary constant-
weight codes with parameters $[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$ and
prove that all their codewords satisfy the condition that $1$, $w$, and
$w^{2}$ appear $2^{k-2}$ times, see Lemmas 3, 6 and 9. In addition, we also
obtain two classes of derived codes and determine their weight distributions,
see Lemma 5.

  2. 2.

We propose an anticode construction of optimal additive codes based on a
combinatorial approach, see Lemma 8 and Theorem 1. With the help of an
equivalent division method, we improve the parameters of the resulting
additive codes, see Theorem 2. The weight distributions of related optimal
additive codes are also completely determined.

  3. 3.

We provide a generalized Construction X. By choosing a special auxiliary
$[n,k^{\prime},d]_{4}$ code and combining it with two $[n_{1},k,d_{1}]_{4}$
and $[n_{2},k,d_{2}]_{4}$ codes, an additive
$[n_{1}+n_{2}+n,k,>d_{1}+d_{2}+d]_{4}$ code can be generated, as shown in
Theorem 3. In addition to directly constructing optimal additive codes, it can
also establish a connection between optimal additive codes with non-integer
dimensions and linear codes, see Lemma 14.

  4. 4.

We give a method to determine the existence of quaternary additive codes using
binary linear codes (Lemma 15). A tighter bound concerning the Griesmer bound
for quaternary additive codes with short lengths is obtained. By the optimal
additive codes obtained in this paper, we ultimately determine optimal
additive $[n,3.5,n-t]_{4}$ codes for all $t$ with variable $n$, except for
$t=6,7,12$, see Lemma 16 and Table II.

This paper is organized as follows. In Section II, we recall some preliminary
results needed in this paper. Sections III and IV are devoted to additive
constant-weight codes and related anticode construction. Section V presents a
generalized Construction X. In Section VI, we determine the parameters of all
the quaternary 3.5-dimensional GPO and PO codes, except for three. Finally,
Section VII concludes this paper.

##  II Preliminaries

This section presents the fundamentals of additive codes. For more details,
refer to the standard textbooks [33, 34, 35].

Denote the finite fields of order $2$ and $4$ by respectively
$F_{2}=\\{0,1\\}$ and $F_{4}=\\{0,1,w,w^{2}\\}$ with $w^{2}+w+1=0$. For
$\bm{u}=(u_{0},\ldots,u_{n-1})\in F_{2}^{n}$, the Hamming weight of $\bm{u}$
is $\mathbf{w}(\bm{u})=|\left\\{i\mid u_{i}\neq 0,0\leq i\leq n-1\right\\}|$.
For $\bm{v}=(v_{0},\ldots,v_{2n-1})\in F_{2}^{2n}$, the symplectic weight of
$\bm{v}$ is
$\mathbf{w}_{s}(\bm{v})=|\left\\{i\mid(v_{i},v_{n+i})\neq(0,0),0\leq i\leq
n-1\right\\}|$. Let $q=2$ or $4$, a code $C_{l}$ is linear over $F_{q}$ if it
is a linear subspace of $F_{q}^{n}$; if such a code has dimension $k$, minimum
Hamming distance $d$, then $C_{l}$ is denoted by $[n,k,d]^{l}_{q}$. If $C_{s}$
is a $[2n,k]_{2}^{l}$ code with the minimum symplectic distance $d_{s}$, then
$C_{s}$ is denoted as $[2n,k,d_{s}]^{s}_{2}$. An additive $(n,K,d)_{4}$ code
$C_{a}$ is an additive subgroup of $F_{4}^{n}$ with $K$ elements whose minimum
Hamming weight is $d$, which is an $F_{2}$-linear $F_{4}$-code. Writing
dimension $k=\log_{4}K$, then $C_{a}$ also can be denoted as $[n,k,d]_{4}$,
where $k$ is not necessarily integer. The ratio $\frac{k}{n}$ of a code is its
information rate.

Define a map $\Phi$ from $F_{2}^{2n}$ to $F_{4}^{n}$ as
$\Phi(\bm{v})=(v_{0}+w\cdot v_{n},v_{1}+w\cdot v_{n+1},\cdots,v_{n-1}+w\cdot
v_{2n-1})$, where $w$ generates $F_{4}$ over $F_{2}$. Then it is easy to see
that $\Phi$ is a bijection and
$\mathbf{w}(\Phi(\bm{v}))=\mathbf{w}_{s}(\bm{v})$. Let $G_{s}$ be a generator
matrix of $C_{s}$. Then, $\Phi(G_{s})$ generates an additive code $C_{a}$ with
parameters $[n,\frac{k}{2},d_{s}]_{4}$. The additive form of $F_{4}$-vector
$\bm{m}=(m_{0},m_{1},\ldots,m_{k-1})^{T}$ is
$(m_{0},w*m_{0},m_{1},w*m_{1},\ldots,m_{k-1},w*m_{k-1})^{T}$. Let
$F_{4_{a}}^{k}$ denote the vector space consisting of all vectors in
$F_{4}^{k}$ written in the additive form. The extended code of $C_{a}$ is
obtained by adding an appropriate extra coordinate to each codeword of $C_{a}$
such that the sum of the coordinates of all extended codewords is zero. The
weight distribution of a code $C_{a}$ is defined by
$1+A_{1}z+A_{2}z^{2}+\cdots+A_{n}z^{n}$, where $A_{i}$ is the number of
codewords of weight $i$, $1\leq i\leq n$. We call $C_{a}$ a $t$-weight code if
the number of nonzero $A_{i}$ in the sequence $(A_{1},A_{2},\cdots,A_{n})$
equals $t$.

A $k$-dimensional vector space over $F_{2}$ can be viewed as $PG(k-1,2)$. In
$PG(k-1,2)$, one-dimensional subspaces are called points, and the two-
dimensional subspaces are called lines. Let $\mathcal{P}$ and $\mathcal{L}$
represent the set of all points and lines in $PG(k-1,2)$, respectively. The
number of points in $PG(k-1,2)$ is $|\mathcal{P}|=2^{k}-1$ and the number of
lines is $|\mathcal{L}|=\frac{|\mathcal{P}|\cdot(|\mathcal{P}|-1)}{6}$. A set
of all points generates the well-known Simplex code with parameters
$[2^{k}-1,k,2^{k-1}]_{2}$, which is also a constant-weight cyclic
code111Binary cyclic Simplex code can be generated by a polynomial with define
set $T=Z_{n}\setminus C_{1}$.. In [10], Blokhuis et al. took quaternary
additive code as multisets of lines in $PG(k-1,2)$, this is, viewed an
$F_{4}$-additive code as an $F_{2}$-code with symplectic weight. Bierbrauer et
al. proposed the parameters
$[\frac{(2^{k}-1)(2^{k-1}-1)}{3},\frac{k}{2},2^{k-2}(2^{k-1}-1)]_{4}$ of
additive Simplex code in [11]. It is easy to see that the additive Simplex
code meets the additive Griesmer bound with equality.

At the end of this section, we introduce a particular lengthening method for
additive codes, called Construction X, which is an efficient technique for
constructing optimal additive codes in [18].

###### Lemma 2

(Additive Construction X, [18]) Assume that $C_{a}$ is an additive
$[n,k_{1},d_{1}]_{4}$ code with subcode $[n,k_{2},d_{2}]_{4}$, where
$d_{2}>d_{1}$. Let $C_{au}$ be an additive $[l,k_{1}-k_{2},\delta]_{4}$ code.
Then, there exists an additive $[n+l,k_{1},\min\\{\delta+d_{1},d_{2}\\}]_{4}$
code.

###### Proof:

Since $C_{a}$ has an $[n,k_{2},d_{2}]_{4}$ subcode, $C_{a}$ can be divided
into two disjoint parts, subcodes $C_{s_{1}}$ and $C_{s_{2}}$ with parameters
$[n,k_{2},d_{2}]_{4}$ and $[n,k_{1}-k_{2},\geq d_{1}]_{4}$, respectively.
Obviously, we have $C_{a}=C_{s_{1}}+C_{s_{2}}$, and it is $C_{s_{2}}$ or the
combination of $C_{s_{1}}$ and $C_{s_{2}}$ decrease the distance of $C_{a}$
from $d_{2}$ be $d_{1}$. An economic approach to augment the distance of
$C_{a}$ to $d_{2}$ is horizontal join an auxiliary code $C_{au}$ with
$C_{s_{2}}$ so that one can make the result code $C_{ax}$ has smaller length
$n+l$.

Let $G_{a}$, $G_{s_{1}}$, $G_{s_{2}}$, $G_{au}$ and $G_{ax}$ denote generator
matrices of $C_{a}$, $C_{s_{1}}$, $C_{s_{2}}$, $C_{au}$ and $C_{ax}$,
respectively. Then, additive Construction X can be described as the following
matrix form:

$G_{ax}=\left(\begin{matrix}G_{s_{1}}&\mathbf{0}_{k_{2}\times l}\\\
G_{s_{2}}&G_{au}\end{matrix}\right),$ (4)

where $\mathbf{0}_{k_{2}\times l}$ is $k_{2}\times l$ zero matrix. Obviously,
$G_{ax}$ generates an additive $[n+l,k_{1},\min\\{\delta+d_{1},d_{2}\\}]_{4}$
code. ∎

##  III Additive symbol equal probability codes

Given $\alpha\in F_{4}$, let $\bm{\alpha_{k,n}}$ denote the $k\times n$ matrix
such that each entry is $\alpha$, and the notation $\bm{\alpha_{k,n}}$ is
simplified as $\bm{\alpha_{n}}$ if $k=1$, and $\bm{\alpha_{k}}^{T}$ if $n=1$.
Let $\bm{c}=(c_{0},c_{1},\cdots,c_{n-1})\in F_{4}^{n}$ and
$\mathbf{w}_{\alpha}(\bm{c})=|\\{i\mid c_{i}=\alpha,0\leq i\leq n-1\\}|$.
Define cyclic shift operator $\tau$ as
$\tau(\bm{c})=(c_{n-1},c_{0},\cdots,c_{n-2})$. Let
$[n,k^{\prime},d^{\prime}]_{4}\subset[n,k,d]_{4}$ denote that the additive
$[n,k,d]_{4}$ code has an $[n,k^{\prime},d^{\prime}]_{4}$ subcode.

###### Definition 1

Let $C_{a}$ be a quaternary additive code. If any codeword $\bm{c}$ of $C_{a}$
satisfies

$\mathbf{w}_{1}(\bm{c})=\mathbf{w}_{w}(\bm{c})=\mathbf{w}_{w^{2}}(\bm{c}),$
(5)

then we call $C_{a}$ an additive symbol equal probability (ASEP) code.

###### Lemma 3

Let $f(x)=f_{0}+\cdots+f_{k-2}x^{k-2}+f_{k-1}x^{k-1}+x^{k}$ be a monic
irreducible polynomial222The property of finite fields guarantees that such
polynomials can be found for any $k$ and $q$. A simple way is to choose the
generator polynomial of $q$-ary ($n-k$)-dimensional cyclic Hamming code. with
degree $k$ over $F_{2}[x]$, and
$\mathcal{S}_{f}=\\{f_{0},f_{1},\ldots,f_{k-1}\\}$. Let
$S_{k}=\begin{pmatrix}\bm{r_{0}}\\\ \bm{r_{1}}\\\ \vdots\\\
\bm{r_{k-1}}\end{pmatrix}$ be a generator matrix of binary $k$-dimensional
Simplex code $C_{S_{k}}$, where $\bm{r_{i}}$ is row vector, $k\geq 3$, $0\leq
i\leq k-1$. Then, we have the following results.

(1) ASEP code $C_{A_{\frac{k}{2}}}$ with parameters
$[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$ can be generated by
$A_{\frac{k}{2}}$.

(2) If $S_{k}$ generates a cyclic Simplex code $C_{S_{k}}$ and satisfies
$\bm{r_{i+1}}=\tau(\bm{r_{i}})$, then $A_{\frac{k}{2}}^{c}$ also generates
$C_{A_{\frac{k}{2}}}$.

$A_{\frac{k}{2}}=\begin{pmatrix}\bm{r_{0}}+w\cdot\bm{r_{1}}\\\
\bm{r_{1}}+w\cdot\bm{r_{2}}\\\ \vdots\\\ \bm{r_{k-2}}+w\cdot\bm{r_{k-1}}\\\
\bm{r_{k-1}}+w\cdot\sum\limits_{i\in\mathcal{S}_{f}}\bm{r_{i}}\end{pmatrix},A_{\frac{k}{2}}^{c}=\begin{pmatrix}\bm{r_{0}}+w\cdot\bm{r_{1}}\\\
\bm{r_{1}}+w\cdot\bm{r_{2}}\\\ \vdots\\\ \bm{r_{k-2}}+w\cdot\bm{r_{k-1}}\\\
\bm{r_{k-1}}+w\cdot\tau(\bm{r_{k-1}})\end{pmatrix}.$ (6)

(3) The ASEP code $C_{A_{\frac{k}{2}}}$ is an additive both GPO and Griesmer
code.

###### Proof:

Let $A_{f}=\begin{pmatrix}&1&&\\\ &&\ddots&\\\ &&&1\\\
f_{0}&f_{1}&\cdots&f_{k-1}\end{pmatrix}$ be a $k\times k$ square matrix over
$F_{q}$, where the empties indicate zero. Let
$G_{s}=(S_{k},A_{f}S_{k})=\left(\begin{array}[]{c|c}\bm{r_{0}}&\bm{r_{1}}\\\
\bm{r_{1}}&\bm{r_{2}}\\\ \vdots\\\
\bm{r_{k-1}}&\sum\limits_{i\in\mathcal{S}_{f}}\bm{r_{i}}\end{array}\right)$.
By Lemma 2.5 in [36], $f(x)$ is a characteristic polynomial of $A_{f}$, so
$Rank(A_{f})=k$ and $A_{f}$ has no eigenvalues in $F_{2}$. Let $C_{s}$ be the
code generated $G_{s}$ and $\bm{c_{I}}$ be an $F_{2}$-vector of length $k$.
Then, $\bm{c}=\bm{c_{I}}(S_{k},A_{f}S_{k})=(\bm{c_{l}},\bm{c_{r}})$ can
represent all codewords of $C_{s}$. Since, $f(x)$ is an irreducible
polynomial, for all nonzero $\bm{c_{I}}$, we have
$\bm{c_{l}}-\bm{c_{r}}=\bm{c_{I}}(I_{k}-A_{f})S_{k}\neq\bm{0}$. Therefore,
$\bm{c_{l}}+\bm{c_{r}}$ is also a nonzero codeword of $C_{S_{k}}$. With Lemma
2.4 in [36],
$\mathbf{w}_{s}(\bm{c})=(\mathbf{w}(\bm{c_{l}})+\mathbf{w}(\bm{c_{r}})+\mathbf{w}(\bm{c_{l}}+\bm{c_{r}}))/2\geq
3\cdot 2^{k-2}$, so $C_{s}$ has parameters $[2^{k+1}-2,\frac{k}{2},3\cdot
2^{k-2}]_{2}^{s}$, i.e., $A_{\frac{k}{2}}=\Phi(G_{s})$ generates an additive
$[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$ code.

In addition, if $C_{S_{k}}$ is a cyclic Simplex code with generator polynomial
$g(x)$, and $S_{k}$ satisfies $\bm{r_{i+1}}=\tau(\bm{r_{i}})$, then the code
generated by $\Phi^{-1}(A_{\frac{k}{2}}^{c})$ can be viewed as a quasi-cyclic
code with generator $(g(x),xg(x))$. By Theorem 1 in [18], $(g(x),xg(x))$
generates a quasi-cyclic code $C_{qc}$ with parameters
$[2^{k+1}-2,\frac{k}{2},3\cdot 2^{k-2}]_{2}^{s}$. Therefore,
$A_{\frac{k}{2}}^{c}$ also generates an additive $[2^{k}-1,\frac{k}{2},3\cdot
2^{k-2}]_{4}$ code.

Furthermore, since $k$-dimensional Simplex code $C_{S_{k}}$ is a constant-
weight linear $[2^{k}-1,k,2^{k-1}]^{l}_{2}$ code , for arbitrary nonzero
codewords $\bm{c_{a}}$ and $\bm{c_{b}}$ ($\bm{c_{a}}\neq\bm{c_{b}}$) in
$C_{S_{k}}$, we have $\mathbf{w}_{0}(\bm{c_{a}}+\bm{c_{b}})=2^{k-1}-1$ and
$\mathbf{w}_{1}(\bm{c_{a}}+\bm{c_{b}})=2^{k-1}$. Obviously, both the left and
right sides of $\Phi^{-1}(A_{\frac{k}{2}})$ and
$\Phi^{-1}(A_{\frac{k}{2}}^{c})$ can generate $C_{S_{k}}$. Also, the two sides
of their nonzero codewords are always different. Therefore,
$\bm{c_{a}}+w\bm{c_{b}}$ can represent all codewords of $C_{A_{\frac{k}{2}}}$,
and we have

$\left\\{\begin{array}[]{l}\mathbf{w}_{0}(\bm{c_{a}}+w\bm{c_{b}})=2^{k-2}-1.\\\
\mathbf{w}_{0}(\bm{c_{a}}+w\bm{c_{b}})+\mathbf{w}_{w}(\bm{c_{a}}+w\bm{c_{b}})=2^{k-1}-1.\\\
\mathbf{w}_{1}(\bm{c_{a}}+w\bm{c_{b}})+\mathbf{w}_{w}(\bm{c_{a}}+w\bm{c_{b}})=2^{k-1}.\\\
\mathbf{w}_{w}(\bm{c_{a}}+w\bm{c_{b}})+\mathbf{w}_{w^{2}}(\bm{c_{a}}+w\bm{c_{b}})=2^{k-1}.\end{array}\right.\Rightarrow\left\\{\begin{array}[]{l}\mathbf{w}_{1}(\bm{c_{a}}+w\bm{c_{b}})=2^{k-2}.\\\
\mathbf{w}_{w}(\bm{c_{a}}+w\bm{c_{b}})=2^{k-2}.\\\
\mathbf{w}_{w^{2}}(\bm{c_{a}}+w\bm{c_{b}})=2^{k-2}.\end{array}\right.$ (7)

Thus, the additive $[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$ codes generated
by $A_{\frac{k}{2}}$ and $A_{\frac{k}{2}}^{c}$ are both ASEP codes.

Finally, since $g(k,3\cdot 2^{k-1})=3(2^{k}-1)$ and $g(k,3\cdot
2^{k-1}+2)=3\cdot 2^{k}+k-2>3\cdot 2^{k}$, by Lemma 1, additive
$[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$ code is an additive both GPO and
Griesmer code. ∎

###### Remark 1

Remarkably, Bierbrauer et al. [37] also recently investigated geometrically
optimal additive codes with parameters $[2^{k}-1,\frac{k}{2},3\cdot
2^{k-2}]_{4}$ and proved they are 3-cover. Here, Lemma 3 can be regarded as
another construction of these optimal additive codes from a different point of
view.

###### Example 1

When $k=3$, we can easily get a 1.5-dimensional ASEP code $C_{A_{1.5}}$ from
3-dimensional Simplex code by Lemma 3. Here, we show the details.

Firstly, the generator matrix $S_{3}$ of binary $3$-dimensional Simplex code
may be chosen as

$S_{3}=\left(\begin{array}[]{ccccccc}1&0&0&1&1&0&1\\\ 0&1&0&1&0&1&1\\\
0&0&1&0&1&1&1\end{array}\right).$ (8)

By (1) of Lemma 3, we can choose a monic irreducible polynomial
$f(x)=x^{3}+x+1$, which corresponding a non-singular matrix
$A_{f}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 1&1&0\end{array}\right)$.
$A_{f}$ has no eigenvalues in $F_{2}$. Then, we have

$A_{1.5}=\Phi(S,A_{f}S)=\left(\begin{array}[]{ccccccc}1&w&0&w^{2}&1&w&w^{2}\\\
0&1&w&1&w&w^{2}&w^{2}\\\ w&w&1&0&w^{2}&w^{2}&1\end{array}\right).$ (9)

It is easy to verify that $A_{1.5}$ generates an ASEP code $C_{1.5}$ with
parameters $[7,1.5,6]_{4}$.

For ASEP Griesmer codes with integer dimensions, we prefer a more
straightforward approach to construct them, as the following lemma.

###### Lemma 4

Let $G_{l}$ be a generator matrix of quaternary $l$-dimensional Simplex code,
where $l\geq 2$. Then, the following matrix $A_{l}$ generates an ASEP Griesmer
code $C_{A_{l}}$ with parameters $[2^{2l}-1,l,3\cdot 2^{2l-2}]_{4}$.

$A_{l}=\begin{pmatrix}G_{l}&wG_{l}&w^{2}G_{l}\\\
wG_{l}&w^{2}G_{l}&G_{l}\end{pmatrix}.$ (10)

###### Proof:

This lemma follows from the properties of Simplex codes. ∎

For convenience, in the paper, we agree that the parameters of the ASEP codes
are all $[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$, and denote them as
$C_{A_{\frac{k}{2}}}$, and default its generator matrix as $A_{\frac{k}{2}}$.

###### Lemma 5

Let the symbols be the same as above. Then, the following results hold.

(1) Verticaljoin $\bm{0_{2^{k}-1}}$, $\bm{1_{2^{k}-1}}$ and $\bm{w_{2^{k}-1}}$
with the generator matrix of $C_{A_{\frac{k}{2}}}$ yields an additive Griesmer
code $C_{Au_{\frac{k}{2}}}$ with parameters $[2^{k}-1,\frac{k}{2}+1,3\cdot
2^{k-2}-1]_{4}$, and there is

$[2^{k}-1,1,2^{k}-1]_{4}\subset[2^{k}-1,\frac{k}{2}+1,3\cdot 2^{k-2}-1]_{4};$
(11)

(2) $C_{Au_{\frac{k}{2}}}$ is a three-weight code with weight distribution
$1+(2^{k+2}-2^{k}-3)\cdot z^{3\cdot 2^{k-2}-1}+(2^{k}-1)\cdot z^{3\cdot
2^{k-2}}+3z^{2^{k}-1}$;

(3) Extending $C_{Au_{\frac{k}{2}}}$ gives an additive both GPO and Griesmer
code $C_{Aue_{\frac{k}{2}}}$ with parameters $[2^{k},\frac{k}{2}+1,3\cdot
2^{k-2}]_{4}$, there also exists

$[2^{k},1,2^{k}]_{4}\subset[2^{k},\frac{k}{2}+1,3\cdot 2^{k-2}]_{4};$ (12)

(4) $C_{Aue_{\frac{k}{2}}}$ is a two-weight code with weight distribution
$1+(2^{k+2}-4)\cdot z^{3\cdot 2^{k-2}}+3z^{2^{k}}$.

###### Proof:

For (1), since $C_{A_{\frac{k}{2}}}$ is an ASEP code, adding
$\bm{0_{2^{k}-1}}$ to $C_{A_{\frac{k}{2}}}$ makes no difference, and for given
any codeword $\bm{c}$ of $C_{A_{\frac{k}{2}}}$ and nonzero element $\beta$ in
$F_{4}$, we have $\mathbf{w}_{0}(\bm{c}+\bm{\beta_{2^{k}-1}})=2^{k-2}$ and

$S=\\{\mathbf{w}_{1}(\bm{c}+\bm{\beta_{2^{k}-1}}),\mathbf{w}_{w}(\bm{c}+\bm{\beta_{2^{k}-1}}),\mathbf{w}_{w^{2}}(\bm{c}+\bm{\beta_{2^{k}-1}})\\}=\\{2^{k-2}-1,2^{k-2}\\}.$
(13)

Furthermore, there is only one value of $2^{k}-1$ in
$\mathbf{w}_{1}(\bm{c}+\bm{\beta_{2^{k}-1}})$,
$\mathbf{w}_{w}(\bm{c}+\bm{\beta_{2^{k}-1}})$ and
$\mathbf{w}_{w^{2}}(\bm{c}+\bm{\beta_{2^{k}-1}})$. Thus,
$d_{H}(C_{Au_{\frac{k}{2}}})\geq 3\cdot 2^{k-2}-1$. Any basis of
$C_{A_{\frac{k}{2}}}$ can not span $\bm{1_{2^{k}-1}}$, $\bm{w_{2^{k}-1}}$ and
$\bm{w^{2}_{2^{k}-1}}$, so the dimension of $C_{Au_{\frac{k}{2}}}$ is
$\frac{k}{2}+1$. Since $g(k+2,3\cdot 2^{k-1}-2)=3(2^{k}-1)$,
$[2^{k}-1,\frac{k}{2}+1,3\cdot 2^{k-2}-1]_{4}$ code is an additive Griesmer
code.

For (2), since $C_{Au_{\frac{k}{2}}}$ has two disjoint subcodes
$[2^{k}-1,1,2^{k}-1]_{4}$ and $[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$,
respectively, all the codewords of weight $3\cdot 2^{k-2}-1$ are a combination
of codewords of these two subcodes. Therefore, $C_{Au_{\frac{k}{2}}}$ has
weight distribution $1+(2^{k+2}-2^{k}-3)\cdot z^{3\cdot
2^{k-2}-1}+(2^{k}-1)\cdot z^{3\cdot 2^{k-2}}+3z^{2^{k}-1}$.

For (3), for the reason that there is $[2^{k}-1,\frac{k}{2},3\cdot
2^{k-2}]_{4}\subset[2^{k}-1,\frac{k}{2}+1,3\cdot 2^{k-2}-1]_{4}$, by extending
additive code $[2^{k}-1,1,2^{k}-1]_{4}$ we can get a
$[2^{k},\frac{k}{2}+1,3\cdot 2^{k-2}]_{4}$ code. Since $g(k+2,3\cdot
2^{k-1})=3\cdot 2^{k}$ and $g(k+2,3\cdot 2^{k-1}+2)=3\cdot 2^{k}+k+1>3\cdot
2^{k}+3$, $[2^{k},\frac{k}{2}+1,3\cdot 2^{k-2}]_{4}$ code is an additive both
GPO and Griesmer code.

For (4), since all combinations of the codewords in $[2^{k},1,2^{k}]_{4}$ code
and $[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}$ code give codewords of weight
$3\cdot 2^{k-2}$, $C_{Aue_{\frac{k}{2}}}$ is a two-weight code with weight
distribution $1+(2^{k+2}-4)\cdot z^{3\cdot 2^{k-2}}+3z^{2^{k}}$. ∎

Based on Lemma 5, we give the following method to construct high-dimensional
ASEP codes from low-dimensional ASEP codes iteratively.

###### Lemma 6

Let the symbols be the same as above. Let $C_{A_{\frac{k}{2}}}$ be a
$\frac{k}{2}$-dimensional ASEP code. Then the ($\frac{k}{2}+1$)-dimensional
ASEP code $C_{A_{\frac{k}{2}+1}}$ can be generated by

$A_{\frac{k}{2}+1}=\begin{pmatrix}A_{\frac{k}{2}}&A_{\frac{k}{2}}&A_{\frac{k}{2}}&A_{\frac{k}{2}}&\bm{0_{k\times
1}}&\bm{0_{k\times 1}}&\bm{0_{k\times 1}}\\\
\bm{0_{1,2^{k}-1}}&\bm{1_{1,2^{k}-1}}&\bm{w_{1,2^{k}-1}}&\bm{w^{2}_{1,2^{k}-1}}&1&w&w^{2}\\\
\bm{0_{1,2^{k}-1}}&\bm{w_{1,2^{k}-1}}&\bm{w^{2}_{1,2^{k}-1}}&\bm{1_{1,2^{k}-1}}&w&w^{2}&1\end{pmatrix}.$
(14)

###### Proof:

The matrix $A_{\frac{k}{2}+1}$ can be divided into four parts as follows:

$A_{\frac{k}{2}+1}=\left(\begin{array}[]{c|cc|cc|cc}A_{\frac{k}{2}}&A_{\frac{k}{2}}&\bm{0_{k\times
1}}&A_{\frac{k}{2}}&\bm{0_{k\times 1}}&A_{\frac{k}{2}}&\bm{0_{k\times 1}}\\\
\bm{0_{1,2^{k}-1}}&\bm{1_{1,2^{k}-1}}&1&\bm{w_{1,2^{k}-1}}&w&\bm{w^{2}_{1,2^{k}-1}}&w^{2}\\\
\bm{0_{1,2^{k}-1}}&\bm{w_{1,2^{k}-1}}&w&\bm{w^{2}_{1,2^{k}-1}}&w^{2}&\bm{1_{1,2^{k}-1}}&1\end{array}\right).$
(15)

By Lemma 5, the code $C_{Aue_{\frac{k}{2}}}$ generated by

$\begin{pmatrix}A_{\frac{k}{2}}&\bm{0_{k,1}}\\\ \bm{\beta_{2^{k}-1}}&\beta\\\
w\cdot\bm{\beta_{2^{k}-1}}&w\cdot\beta\end{pmatrix}$ (16)

is a two-weight additive code with parameters $[2^{k},\frac{k}{2}+1,3\cdot
2^{k-2}]_{4}$.

Therefore,
$A^{\prime}_{\frac{k}{2}+1}=\left(\begin{array}[]{cc|cc|cc}A_{\frac{k}{2}}&\bm{0_{k,1}}&A_{\frac{k}{2}}&\bm{0_{k,1}}&A_{\frac{k}{2}}&\bm{0_{k,1}}\\\
\bm{1_{1,2^{k}-1}}&1&\bm{w_{1,2^{k}-1}}&w&\bm{w^{2}_{1,2^{k}-1}}&w^{2}\\\
\bm{w_{1,2^{k}-1}}&w&\bm{w^{2}_{1,2^{k}-1}}&w^{2}&\bm{1_{1,2^{k}-1}}&1\end{array}\right)$
generates a two-weight additive code $C_{A_{\frac{k}{2}+1}}^{\prime}$ with
parameters $[3\cdot 2^{k},\frac{k}{2}+1,9\cdot 2^{k-2}]_{4}$. For any nonzero
codeword $\bm{c^{\prime}}$ in $C_{A_{\frac{k}{2}+1}}^{\prime}$, there is
$\mathbf{w}_{0}(\bm{c^{\prime}})=\mathbf{w}_{1}(\bm{c^{\prime}})=\mathbf{w}_{w}(\bm{c^{\prime}})=\mathbf{w}_{w^{2}}(\bm{c^{\prime}})=3\cdot
2^{k-2}$ or
$\mathbf{w}_{1}(\bm{c^{\prime}})=\mathbf{w}_{w}(\bm{c^{\prime}})=\mathbf{w}_{w^{2}}(\bm{c^{\prime}})=2^{k}$.
Thus, $A_{\frac{k}{2}+1}$ can generate an additive code with parameters
$[2^{k+2}-1,\frac{k}{2}+1,3\cdot 2^{k}]_{4}$, which satisfies for any nonzero
codeword$\bm{c^{\prime\prime}}$ in $C_{A_{\frac{k}{2}+1}}$,
$\mathbf{w}_{1}(\bm{c^{\prime\prime}})=\mathbf{w}_{w}(\bm{c^{\prime\prime}})=\mathbf{w}_{w^{2}}(\bm{c^{\prime\prime}})=2^{k}$.
Therefore, it is indeed ($\frac{k}{2}+1$)-dimensional ASEP code
$C_{A_{\frac{k}{2}+1}}$. ∎

By Lemma 6, we also can iteratively construct $C_{A_{\frac{k}{2}}}$ from
$C_{1.5}$ for $k>3$ and $k$ is odd, such that it has an ordered matrix
structure.

###### Lemma 7

If there exists an additive code $C_{a}$ with parameters
$[n,\frac{k}{2},d]_{4}$, then there also exists an additive code with
parameters

$[s\cdot 2^{k}+n,\frac{k}{2}+1,3s\cdot 2^{k-2}+d]_{4},$ (17)

where $s\geq\lceil\frac{d}{2^{k-2}}\rceil$.

###### Proof:

According to (3) of Lemma 5, we have an special additive
$[2^{k},\frac{k}{2}+1,3\cdot 2^{k-2}]_{4}$ code $C_{Aue_{\frac{k}{2}}}$, which
has a $[2^{k},1,2^{k}]_{4}$ subcode. Therefore, the juxtaposition of $s$
generator matrix of $C_{Aue_{\frac{k}{2}}}$ produces additive $[s\cdot
2^{k},\frac{k}{2}+1,3s\cdot 2^{k-2}]_{4}$ code with an $[s\cdot 2^{k},1,s\cdot
2^{k}]_{4}$ subcode. Furthermore, since $s\geq\lceil\frac{d}{2^{k-2}}\rceil$,
using $C_{a}$ as auxiliary code, we can get an additive $[s\cdot
2^{k}+n,\frac{k}{2}+1,3s\cdot 2^{k-2}+d]_{4}$ code by Lemma 2. ∎

Lemma 7 is also an effective method for constructing high-dimensional additive
codes from low-dimensional additive codes. Lemma 6 is a special case of Lemma
7.

##  IV Additive anticode construction

Let $C_{a}$ be an additive code with generator matrix $A$. We define generator
multiset $\mathcal{L}$ of $C_{a}$ as a set consisting of column vectors of
$A$. Define $d_{max}(C_{a})$ as the maximum Hamming weight of $C_{a}$. For a
$k\times n$ matrix $A$ with no zero column, we define
$A^{\circ}=(A\mid\bm{0_{k}}^{T})$. For any matrix with the type of
$B=(A\mid\bm{0_{k}}^{T})$, we define $(B)^{\ominus}=A$. Throughout this
section, we let $k=k_{1}+k_{2}$ such that $k_{1}\geq 3$ and $k_{2}\geq 4$ are
respective odd and even integers.

###### Lemma 8

(Additive Anticode Construction) Let $C_{a_{1}}$ and $C_{a_{2}}$ be two
additive codes with parameters $[n_{1},k_{1},d_{1}]_{4}$ and
$[n_{2},k_{2}]_{4}$, respectively. If their generator sets
$\mathcal{L}_{a_{1}}$ and $\mathcal{L}_{a_{2}}$ satisfy
$\mathcal{L}_{a_{2}}\subset\mathcal{L}_{a_{1}}$, then we call $C_{a_{2}}$ as
an additive anticode of $C_{a_{1}}$.
$\mathcal{L}_{a_{1}}\setminus\mathcal{L}_{a_{2}}$ generates an additive
$[n_{1}-n_{2},\leq k_{1},\geq d_{1}-d_{max}(C_{a_{2}})]_{4}$ code.

###### Proof:

Since $\mathcal{L}_{a_{2}}\subset\mathcal{L}_{a_{1}}$, we can reorder the
columns of generator matrix of $C_{a_{1}}$ such that any codeword
$\bm{c}=(\bm{c}_{1},\bm{c}_{2})$ of $C_{a_{1}}$ satisfy $\bm{c}_{2}\in
C_{a_{1}}$. Then, $\mathbf{w}(\bm{c}_{1})\geq d_{1}-d_{max}(C_{a_{2}})$.
Puncture the coordinates of $\bm{c}_{2}$ gives a result code with minimum
distance at least $d_{1}-d_{max}(C_{a_{2}})$. In addition, the dimension of
the result code does not exceed $k_{1}$. ∎

Next, we show the specific procedure for constructing optimal additive codes
using ASEP codes by Lemma 8.

###### Definition 2

Let $A=(\bm{\alpha_{1}},\bm{\alpha_{2}},\cdots,\bm{\alpha_{n_{1}}})$,
$B=(\bm{\beta_{1}},\bm{\beta_{2}},\cdots,\bm{\beta_{n_{2}}})$, where
$\bm{\alpha_{i}}$ and $\bm{\beta_{j}}$, $1\leq i\leq n_{1}$, $1\leq j\leq
n_{2}$, are column vectors. Define

$A\star
B=\left(\begin{array}[]{ccc|ccc|c|ccc}\bm{\alpha}_{1}&\cdots&\bm{\alpha_{1}}&\bm{\alpha_{2}}&\cdots&\bm{\alpha_{2}}&\cdots&\bm{\alpha_{n_{1}}}&\cdots&\bm{\alpha_{n_{1}}}\\\
\bm{\beta_{1}}&\cdots&\bm{\beta_{n_{2}}}&\bm{\beta_{1}}&\cdots&\bm{\beta_{n_{2}}}&\cdots&\bm{\beta_{1}}&\cdots&\bm{\beta_{n_{2}}}\end{array}\right).$
(18)

###### Lemma 9

Let $C_{A_{\frac{k_{1}}{2}}}$ and $C_{A_{\frac{k_{2}}{2}}}$ be ASEP codes with
generator matrices $A_{\frac{k_{1}}{2}}$ and $A_{\frac{k_{2}}{2}}$, generated
by Lemmas 6 and 4, respectively. Then

$A_{\frac{k}{2}}=(A_{\frac{k_{1}}{2}}^{\circ}\star
A_{\frac{k_{2}}{2}}^{\circ})^{\ominus}\cong\left(A_{\frac{k_{1}}{2}}\star
A_{\frac{k_{2}}{2}}\begin{array}[]{|cc}A_{\frac{k_{1}}{2}}&\bm{0_{k_{1},2^{k_{2}}-1}}\\\
\bm{0_{k_{2},2^{k_{1}}-1}}&A_{\frac{k_{2}}{2}}\end{array}\right).$ (19)

generates $\frac{k}{2}$-dimensional ASEP code $C_{A_{\frac{k}{2}}}$.

###### Proof:

In the case of using $C_{A_{1.5}}$ as initial ASEP code, the iteratively
constructed $A_{\frac{k}{2}}$ satisfies that lower $k_{2}$-rows of
$A_{\frac{k}{2}}$ in Equation (14) is that all vectors in
$F_{4_{a}}^{\frac{k_{2}}{2}}$ repeat $2^{k_{1}}$ times, which can be written
as

$A_{\frac{k}{2}}^{\circ}=\left(\begin{array}[]{cccc}A_{\frac{k_{1}}{2}}^{\circ}&A_{\frac{k_{1}}{2}}^{\circ}&\ldots&A_{\frac{k_{1}}{2}}^{\circ}\\\
\bm{0_{k_{2},2^{k_{1}}}}&\bm{\mathsf{v}^{(1)}_{k_{2},2^{k_{1}}}}&\ldots&\bm{\mathsf{v}^{(2^{k_{2}}-1)}_{k_{2},2^{k_{1}}}}\end{array}\right),$
(20)

where $\bm{\mathsf{v}^{(i)}_{k_{2},2^{k_{1}}}}$, $1\leq i\leq 2^{k_{2}}-1$,
denotes the $k_{2}\times 2^{k_{1}}$ matrix generated by repeating the $i$-th
nonzero vector in $F_{4_{a}}^{\frac{k_{2}}{2}}$ $2^{k_{1}}$ times.

In addition, ASEP codes with integer dimensions generated by Equation (10) are
indeed all nonzero vectors in in $F_{4_{a}}^{\frac{k_{2}}{2}}$. Therefore, it
is easy to verify that rearranging the columns in Equation (14) yields
Equation (19). ∎

Let $C_{\frac{k}{2}}$ be the ASEP code determined by Equation (19), and denote
its generator set as $\mathcal{L}_{\frac{k}{2}}$. Let

$A_{\frac{k_{1}^{\prime}}{2}}=\left(\begin{array}[]{c}A_{\frac{k_{1}}{2}}\\\
\bm{0_{k_{2}\times(2^{k_{1}}-1)}}\end{array}\right),A_{\frac{k_{2}^{\prime}}{2}}=\left(\begin{array}[]{c}\bm{0_{k_{1}\times(2^{k_{2}}-1)}}\\\
A_{\frac{k_{2}}{2}}\end{array}\right),$ (21)

where $A_{\frac{k_{1}}{2}}$ and $A_{\frac{k_{2}}{2}}$ generated by Lemmas 6
and 4, respectively. $\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$ and
$\mathcal{L}_{\frac{k_{2}^{\prime}}{2}}$ are generator sets of
$A_{\frac{k_{1}^{\prime}}{2}}$ and $A_{\frac{k_{2}^{\prime}}{2}}$,
respectively. Obviously, $\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$ and
$\mathcal{L}_{\frac{k_{2}^{\prime}}{2}}$ are two disjoint subsets of
$\mathcal{L}_{\frac{k}{2}}$.

Therefore, the additive codes $C_{\frac{k_{1}^{\prime}}{2}}$ and
$C_{\frac{k_{2}^{\prime}}{2}}$ determined by $A_{\frac{k_{1}^{\prime}}{2}}$
and $A_{\frac{k_{2}^{\prime}}{2}}$ are anticodes of $C_{\frac{k}{2}}$, and
their maximum weight are separately $3\cdot 2^{k_{1}-2}$ and $3\cdot
2^{k_{2}-2}$.

###### Theorem 1

Let the symbols be the same as above. Then, we have the following results.

(1) $\mathcal{L}_{\frac{k}{2}}\setminus\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$
generates an additive two-weight code with parameters
$[2^{k}-2^{k_{1}},\frac{k}{2},3\cdot 2^{k-2}-3\cdot 2^{k_{1}-2}]_{4}$, which
has the weight distribution $1+(2^{k}-2^{k_{2}})\cdot z^{3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{3\cdot 2^{k-2}}$;

(2) If $k_{1}=3$, then $[2^{k}-2^{k_{1}},\frac{k}{2},3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}]_{4}$ code is GDO; if $k_{1}\geq 5$, it is GPO.

(3)
$\mathcal{L}_{\frac{k}{2}}\setminus(\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}+\mathcal{L}_{\frac{k_{2}^{\prime}}{2}})$
generates an additive GDO three-weight code with parameters
$[2^{k}-2^{k_{1}}-2^{k_{2}}+1,\frac{k}{2},3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}-3\cdot 2^{k_{2}-2}]_{4}$, which has the weight distribution
$1+(2^{k}-2^{k_{1}}-2^{k_{2}})\cdot z^{3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}-3\cdot 2^{k_{2}-2}}+(2^{k_{1}}-1)\cdot z^{3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{3\cdot 2^{k-2}-3\cdot 2^{k_{2}-2}}$.

(4) $[2^{k}-2^{k_{1}}-2^{k_{2}}+1,\frac{k}{2},3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}-3\cdot 2^{k_{2}-2}]_{4}$ code is GDO if $|k_{1}-k_{2}|=1$ and
$9\leq k\leq 15$, or $|k_{1}-k_{2}|\geq 3$ and $\min\\{k_{1},k_{2}\\}\leq 4$;
else if $|k_{1}-k_{2}|=1$ and $k\geq 17$, or $|k_{1}-k_{2}|\geq 3$ and
$\min\\{k_{1},k_{2}\\}>4$, it is GPO.

###### Proof:

In accordance with the foregoing, the corresponding matrix
$A_{\frac{k}{2}\setminus\frac{k_{1}^{\prime}}{2}}$ of
$\mathcal{L}_{\frac{k}{2}}\setminus\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$ can
be written as the following form:

$A_{\frac{k}{2}\setminus\frac{k_{1}^{\prime}}{2}}=\left(A_{\frac{k_{1}}{2}}\star
A_{\frac{k_{2}}{2}}\begin{array}[]{|c}\bm{0_{k_{1},2^{k_{2}}-1}}\\\
A_{\frac{k_{2}}{2}}\end{array}\right).$ (22)

Denote the additive codes generated by the upper $k_{1}$ rows and lower
$k_{2}$ rows of $A_{\frac{k}{2}\setminus\frac{k_{1}^{\prime}}{2}}$ as $C_{up}$
and $C_{down}$. Then, by the properties of
$A_{\frac{k}{2}\setminus\frac{k_{1}^{\prime}}{2}}$, for all nonzero codewords
$\bm{c_{1}}\in C_{up}$ and $\bm{c_{2}}\in C_{down}$, there are
$\mathbf{w}(\bm{c_{1}})=\mathbf{w}(\bm{c_{1}}+\bm{c_{2}})=3\cdot
2^{k-2}-3\cdot 2^{k_{1}-2}$ and $\mathbf{w}(\bm{c_{2}})=3\cdot 2^{k-2}$.
Therefore, the intersection of $C_{up}$ and $C_{down}$ is a zero codeword, it
is easy to conclude that
$\mathcal{L}_{\frac{k}{2}}\setminus\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$
generates a two-weight $[2^{k}-2^{k_{1}},\frac{k}{2},3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}]_{4}$ code with weight distribution $1+(2^{k}-2^{k_{2}})\cdot
z^{3\cdot 2^{k-2}-3\cdot 2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{3\cdot 2^{k-2}}$.
Since $g(k,3\cdot 2^{k-1}-3\cdot 2^{k_{1}-1}+2)=3\cdot 2^{k}-3\cdot
2^{k_{1}}+k_{1}>3(2^{k}-2^{k_{1}})$, this code is GDO for $k_{1}=3$; if
$k_{1}>3$, it is GPO.

Similarly, the corresponding matrix of
$\mathcal{L}_{\frac{k}{2}}\setminus(\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}+\mathcal{L}_{\frac{k_{2}^{\prime}}{2}})$
is $A_{\frac{k_{1}}{2}}\star A_{\frac{k_{2}}{2}}$, which also can be divided
to up and down two parts. Let $C_{up}^{\prime}$ and $C_{down}^{\prime}$ denote
the additive codes generated by the upper $k_{1}$ rows and lower $k_{2}$ rows
of $A_{\frac{k_{1}}{2}}\star A_{\frac{k_{2}}{2}}$, respectively. Then, for all
nonzero codewords $\bm{c_{1}^{\prime}}\in C_{up}^{\prime}$ and
$\bm{c_{2}^{\prime}}\in C_{down}^{\prime}$, there are
$\mathbf{w}(\bm{c_{1}^{\prime}})=3\cdot 2^{k-2}-3\cdot 2^{k_{1}-2}$,
$\mathbf{w}(\bm{c_{2}^{\prime}})=3\cdot 2^{k-2}-3\cdot 2^{k_{2}-2}$ and
$\mathbf{w}(\bm{c_{1}^{\prime}}+\bm{c_{2}^{\prime}})=3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}-3\cdot 2^{k_{2}-2}$. Thus, the intersection of $C_{up}^{\prime}$
and $C_{down}^{\prime}$ is also only zero codeword,
$\mathcal{L}_{\frac{k}{2}}\setminus(\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}+\mathcal{L}_{\frac{k_{2}^{\prime}}{2}})$
generates a three-weight $[2^{k}-2^{k_{1}}-2^{k_{2}}+1,\frac{k}{2},3\cdot
2^{k-2}-3\cdot 2^{k_{1}-2}-3\cdot 2^{k_{2}-2}]_{4}$ code with weight
distribution $1+(2^{k}-2^{k_{1}}-2^{k_{2}})\cdot z^{3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}-3\cdot 2^{k_{2}-2}}+(2^{k_{1}}-1)\cdot z^{3\cdot 2^{k-2}-3\cdot
2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{3\cdot 2^{k-2}-3\cdot 2^{k_{2}-2}}$.

Since $g(k,3\cdot 2^{k-1}-3\cdot 2^{k_{1}-1}-3\cdot
2^{k_{2}-1}+2)=\left\\{\begin{matrix}3\cdot 2^{k}-3\cdot 2^{k_{1}}-3\cdot
2^{k_{2}}+\min\\{k_{1},k_{2}\\},&|k_{1}-k_{2}|=1,\\\ 3\cdot 2^{k}-3\cdot
2^{k_{1}}-3\cdot 2^{k_{2}}+\min\\{k_{1},k_{2}\\}+2,&|k_{1}-k_{2}|\geq
3,\end{matrix}\right.$, $[2^{k}-2^{k_{1}}-2^{k_{2}}+1,\frac{k}{2},3\cdot
2^{k-2}-3\cdot 2^{k_{1}-2}-3\cdot 2^{k_{2}-2}]_{4}$ code is GDO if
$|k_{1}-k_{2}|=1$ and $9\leq k\leq 15$, or $|k_{1}-k_{2}|\geq 3$ and
$\min\\{k_{1},k_{2}\\}\leq 4$; else if $|k_{1}-k_{2}|=1$ and $k\geq 17$, or
$|k_{1}-k_{2}|\geq 3$ and $\min\\{k_{1},k_{2}\\}>4$, it is GPO. ∎

As agreed upon, $A_{\frac{k_{2}}{2}}$ is generated from Lemma 4, which can be
divided into three equivalent blocks, i.e., $A_{\frac{k_{2}^{\prime}}{2}}$ can
be written as
$A_{\frac{k_{2}^{\prime}}{2}}=\left(\begin{array}[]{c|c|c}\bm{0_{k_{1}\times\frac{2^{k_{2}}-1}{3}}}&\bm{0_{k_{1}\times\frac{2^{k_{2}}-1}{3}}}&\bm{0_{k_{1}\times\frac{2^{k_{2}}-1}{3}}}\\\
G_{\frac{k_{2}}{2}}&wG_{\frac{k_{2}}{2}}&w^{2}G_{\frac{k_{2}}{2}}\\\
wG_{\frac{k_{2}}{2}}&w^{2}G_{\frac{k_{2}}{2}}&G_{\frac{k_{2}}{2}}\end{array}\right)$,
where $G_{\frac{k_{2}}{2}}$ is a generator matrix of
$\frac{k_{2}}{2}$-dimensional quaternary Simplex code. Let
$A_{\frac{k_{2}^{\prime}}{2},m}$ denote the first $m$ blocks of
$A_{\frac{k_{2}^{\prime}}{2}}$, where $1\leq m\leq 3$. Then, we have the
following corollary.

###### Corollary 1

Let the symbols be the same as above. Let
$\mathcal{L}_{\frac{k_{2}^{\prime}}{2},m}$ be the generator set of
$C_{\frac{k_{2}^{\prime}}{2},m}$. Then, the following results hold.

(1)
$\mathcal{L}_{\frac{k}{2}}\setminus\mathcal{L}_{\frac{k_{2}^{\prime}}{2},m}^{\prime}$
generates a two-weight code with parameters
$[2^{k}-1-m\cdot\frac{2^{k_{2}}-1}{3},\frac{k}{2},3\cdot 2^{k-2}-m\cdot
2^{k_{2}-2}]_{4}$, whose weight distribution is $1+(2^{k}-2^{k_{1}})\cdot
z^{3\cdot 2^{k-2}-m\cdot 2^{k_{2}-2}}+(2^{k_{1}}-1)\cdot z^{3\cdot 2^{k-2}}$;

(2) The $[2^{k}-1-m\cdot\frac{2^{k_{2}}-1}{3},\frac{k}{2},3\cdot
2^{k-2}-m\cdot 2^{k_{2}-2}]_{4}$ code is an additive both GPO and Griesmer
code for $m=1$, others are just GPO;

###### Proof:

One can easily conclude that the above constructions hold according to the
method of proving Theorem 1. Moreover, since $g(k,3\cdot 2^{k-1}-m\cdot
2^{k_{2}-1})=3\cdot 2^{k}-m\cdot 2^{k_{2}}+\lceil\frac{m}{2}\rceil-3$ and
$g(k,3\cdot 2^{k-1}-m\cdot 2^{k_{2}-1}+2)=3\cdot 2^{k}-m\cdot
2^{k_{2}}+k_{2}+\lceil\frac{m}{2}\rceil-2>3(2^{k}-m\cdot\frac{2^{k_{2}}-1}{3})$,
if $m=1$, $[2^{k}-1-m\cdot\frac{2^{k_{2}}-1}{3},\frac{k}{2},3\cdot
2^{k-2}-m\cdot 2^{k_{2}-2}]_{4}$ code is both additive GPO and Griesmer code,
otherwise, it is just GPO code. ∎

###### Theorem 2

Let the symbols be the same as above, and then we will have the following
results.

(1) $\mathcal{L}_{\frac{k}{2}}\setminus\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$
can be divided into three equivalent parts, each part generates an additive
two-weight additive code with parameters
$[\frac{2^{k}-2^{k_{1}}}{3},\frac{k}{2},2^{k-2}-2^{k_{1}-2}]_{4}$, whose
weight distribution is $1+(2^{k}-2^{k_{2}})\cdot
z^{2^{k-2}-2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{2^{k-2}}$;

(2)
$\mathcal{L}_{\frac{k}{2}}\setminus(\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}+\mathcal{L}_{\frac{k_{2}^{\prime}}{2}})$
can be divided into three equivalent parts, each part generates a three-weight
additive code with parameters
$[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k}{2},2^{k-2}-2^{k_{1}-2}-2^{k_{2}-2}]_{4}$,
and weight distribution $1+(2^{k}-2^{k_{1}}-2^{k_{2}})\cdot
z^{2^{k-2}-2^{k_{1}-2}-2^{k_{2}-2}}+(2^{k_{1}}-1)\cdot
z^{2^{k-2}-2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{2^{k-2}-2^{k_{2}-2}}$.

(3) The codes in (1) and (2) are both GPO and Griesmer codes.

###### Proof:

Since $\frac{k}{2}$-dimensional ASEP code $C_{A_{\frac{k}{2}}}$ can be
generated by $(A_{\frac{k_{1}}{2}}^{\circ}\star
A_{\frac{k_{2}}{2}}^{\circ})^{\ominus}$, and $A_{\frac{k_{2}}{2}}$ can be
divided into three parts, the generator matrix of
$\mathcal{L}_{\frac{k}{2}}\setminus\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$ can
be written as

$\left(\begin{array}[]{c|c|c}A_{\frac{k_{1}}{2}}^{\circ}\star\begin{pmatrix}G_{\frac{k_{2}}{2}}\\\
w*G_{\frac{k_{2}}{2}}\end{pmatrix}&A_{\frac{k_{1}}{2}}^{\circ}\star\begin{pmatrix}w*G_{\frac{k_{2}}{2}}\\\
w^{2}*G_{\frac{k_{2}}{2}}\end{pmatrix}&A_{\frac{k_{1}}{2}}^{\circ}\star\begin{pmatrix}w^{2}*G_{\frac{k_{2}}{2}}\\\
G_{\frac{k_{2}}{2}}\end{pmatrix}\end{array}\right).$ (23)

Since the additive code generated by
$\begin{pmatrix}w^{2}*G_{\frac{k_{2}}{2}}\\\ G_{\frac{k_{2}}{2}}\end{pmatrix}$
is also a linear code, for all $\alpha$ in $F_{4}^{*}$, the additive codes
generated by $\begin{pmatrix}\alpha\cdot G_{\frac{k_{2}}{2}}\\\ \alpha\cdot
w*G_{\frac{k_{2}}{2}}\end{pmatrix}$ are same. Therefore, the codes generated
by $A_{\frac{k_{1}}{2}}^{\circ}\star\begin{pmatrix}G_{\frac{k_{2}}{2}}\\\
w*G_{\frac{k_{2}}{2}}\end{pmatrix}$ and
$A_{\frac{k_{1}}{2}}^{\circ}\star\begin{pmatrix}\alpha\cdot
G_{\frac{k_{2}}{2}}\\\ \alpha\cdot w*G_{\frac{k_{2}}{2}}\end{pmatrix}$ are the
same. Similarly, the codes generated by
$A_{\frac{k_{1}}{2}}\star\begin{pmatrix}G_{\frac{k_{2}}{2}}\\\
w*G_{\frac{k_{2}}{2}}\end{pmatrix}$ and
$A_{\frac{k_{1}}{2}}\star\begin{pmatrix}\alpha\cdot G_{\frac{k_{2}}{2}}\\\
\alpha\cdot w*G_{\frac{k_{2}}{2}}\end{pmatrix}$ are also the same.

Thus, the additive codes generated by
$\mathcal{L}_{\frac{k}{2}}\setminus\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}$ and
$\mathcal{L}_{\frac{k}{2}}\setminus(\mathcal{L}_{\frac{k_{1}^{\prime}}{2}}+\mathcal{L}_{\frac{k_{2}^{\prime}}{2}})$
both can be divided into three equivalent parts. The result additive codes
have similar parameters and weight distributions with those in Theorem 1, but
only need to replace the lengths and distances by one-third of the original
codes.

Since $g(k,2^{k-1}-2^{k_{1}-1})=2^{k}-2^{k_{1}}$ and
$g(k,2^{k-1}-2^{k_{1}-1}+2)=2^{k}-2^{k_{1}}+k_{1}+1$,
$[\frac{2^{k}-2^{k_{1}}}{3},\frac{k}{2},2^{k-2}-2^{k_{1}-2}]_{4}$ code is both
GPO and Griesmer code. In addition, as
$g(k,2^{k-1}-2^{k_{1}-1}-2^{k_{2}-1})=2^{k}-2^{k_{1}}-2^{k_{2}}+1$ and
$g(k,2^{k-1}-2^{k_{1}-1}-2^{k_{2}-1}+2)=2^{k}-2^{k_{1}}-2^{k_{2}}+\min\\{k_{1},k_{2}\\}+2>2^{k}-2^{k_{1}}-2^{k_{2}}+4$,
$[\frac{2^{k}-2^{k_{1}}}{3},\frac{k}{2},2^{k-2}-2^{k_{1}-2}]_{4}$ code is also
both GPO and Griesmer code. ∎

##  V Generalized construction X

To describe the matrix combination method in this section, we denote by
$A_{[n,k]_{4}}$ or $A_{[n,k,d]_{4}}$ a generator matrix of the additive code
with parameters $[n,k,d]_{4}$.

###### Definition 3

If $C_{a}$ is an additive $[n,k_{1},\delta_{1}]_{4}$ code which has two
disjoint subcodes $C_{s_{1}}$ and $C_{s_{2}}$ with parameters
$[n,k_{1}-k_{2},\delta_{1}]_{4}$ and $[n,k_{2},\delta_{2}]_{4}$, respectively,
satisfying for all nonzero codewords $\bm{c_{1}}$ of $C_{s_{1}}$ and
$\bm{c_{2}}$ of $C_{s_{2}}$,
$\delta_{2}=\mathbf{w}(\bm{c_{1}}+\bm{c_{2}})>\delta_{1}$. Then, we call
$C_{a}$ has an invariant subcode $C_{s_{2}}$ with parameters
$[n,k_{2},\delta_{2}]_{4}$.

###### Theorem 3

(Generalized Construction X) Let $C_{1}$ and $C_{2}$ be additive codes with
parameters $[n_{1},k,d_{1}]_{4}$ and $[n_{2},k,d_{2}]_{4}$, respectively,
satisfying
$[n_{1},k_{1}^{\prime},d_{1}^{\prime}]_{4}\subset[n_{1},k,d_{1}]_{4}$, and
$[n_{2},k_{2}^{\prime},d_{2}^{\prime}]_{4}\subset[n_{2},k,d_{2}]_{4}$. Suppose
$k_{1}=k-k_{1}^{\prime}$, $k_{2}=k-k_{2}^{\prime}$, $k_{1}>k_{2}$,
$\delta_{1}=d_{1}^{\prime}-d_{1}$,
$\delta_{2}=d_{2}^{\prime}-d_{1}+\delta_{1}$. Choosing auxiliary code $C_{au}$
with parameters $[n,k_{1},\delta_{1}]_{4}$, which has an invariant subcode
$C_{isub}$ with parameters $[n,k_{2},\delta_{2}]_{4}$, then there is an
additive code with parameters $[n_{1}+n_{2}+n,k,d_{1}+d_{2}+\delta_{2}]_{4}$.

###### Proof:

Firstly, $C_{1}$ and $C_{au}$ can determine an additive code $C_{X_{1}}$ with
parameters $[n_{1}+n,k,d_{1}+\delta_{1}]_{4}$ by Construction X, whose
generator matrix can be written as $A_{X_{1}}$.

$A_{X_{1}}=\left(\begin{array}[]{c|c}A_{[n_{1},k_{1},d_{1}]_{4}}&A_{[n,k_{1},\delta_{1}]_{4}}\\\
A_{[n_{1},k_{1}^{\prime},d_{1}^{\prime}]_{4}}&\bm{0_{(k-k_{1}),n}}\\\
\end{array}\right)=\left(\begin{array}[]{c|c}A_{[n_{1},k_{1},d_{1}]_{4}}&\begin{matrix}A_{[n,k_{2},\delta_{2}]_{4}}\\\
A_{[n,k_{1}-k_{2},\delta_{1}]_{4}}\end{matrix}\\\
A_{[n_{1},k_{1}^{\prime},d_{1}^{\prime}]_{4}}&\bm{0_{k_{1}^{\prime},n}}\\\
\end{array}\right).$ (24)

Let $C_{X_{1a}}$ and $C_{X_{1b}}$ be the additive codes determined by the
first $k_{2}$ rows of $A_{X_{1}}$ and the others rows, respectively. Since
$C_{au}$ has an invariant subcode $C_{isub}$ with parameters
$[n,k_{2},\delta_{2}]_{4}$ satisfying for any nonzero codeword $\bm{c}$ in
$C_{au}\setminus C_{isub}$, $\mathbf{w}(\bm{c})\geq\delta_{2}$, then for all
nonzero codewords $\bm{c_{1}}$ of $C_{X_{1a}}$ and $\bm{c_{2}}$ of
$C_{X_{1b}}$, we have $\mathbf{w}(\bm{c_{1}}+\bm{c_{2}})\geq
d_{1}+\delta_{2}$. Therefore, $C_{X_{1a}}$ can help to increase the distance
of a $\frac{k_{2}}{2}$-dimensional subspace of $C_{2}$ by
$\delta_{2}-\delta_{1}$. The specific combination method is as following
matrix.

$A_{X_{2}}=\left(\begin{array}[]{c|c|c}A_{[n_{1},k_{1},d_{1}]_{4}}&A_{[n_{2},k_{2},d_{2}]_{4}}&\begin{matrix}A_{[n,k_{2},\delta_{2}]_{4}}\\\
A_{[n,k_{1}-k_{2},\delta_{1}]_{4}}\end{matrix}\\\
A_{[n_{1},k_{1}^{\prime},d_{1}^{\prime}]_{4}}&A_{[n_{2},k_{2}^{\prime},d_{2}^{\prime}]_{4}}&\bm{0_{k_{1}^{\prime},n}}\\\
\end{array}\right).$ (25)

It is easy to judge that $A_{X_{2}}$ can generate an additive code with
parameters $[n_{1}+n_{2}+n,k,d_{1}+d_{2}+\delta_{2}]_{4}$. ∎

Theorem 4 is an enhancement of Construction X. Compared with Construction X,
the present construction enlarges the distance by $\delta_{2}$ rather than
$\delta_{1}$. For convenience, we also have the following corollary.

###### Corollary 2

Let $C_{1}$ and $C_{2}$ be additive codes with parameters
$[n_{1},k,d_{1}]_{4}$ and $[n_{2},k,d_{2}]_{4}$, respectively. If $C_{1}$ has
a subcode with parameters $[n_{1},k_{1}^{\prime},d_{1}^{\prime}]_{4}$ and
$C_{2}$ has an invariant subcode with parameters
$[n,k-k_{1}^{\prime},d_{2}+d_{1}^{\prime}-d_{1}]_{4}$, then there exists an
additive code with parameters $[n_{1}+n_{2},k,d_{1}^{\prime}+d_{2}]_{4}$.

###### Proof:

This corollary holds from the procedure of proving Theorem 3. ∎

Additive codes with invariant subcodes are essential in implementing the
generalized Construction X. In the following, we focus on constructing optimal
additive codes with invariant subcodes.

###### Lemma 10

Let $k\geq 5$ be an odd number. Then, there exists an additive GPO
$[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}$ code with weight distribution
$1+(2^{k-1}-1)z^{2^{k-2}}+2^{k-1}z^{2^{k-2}+1}$, which has an invariant
subcode with parameters $[\frac{2^{k}+1}{3},0.5,2^{k-2}+1]_{4}$.

###### Proof:

Here, we use an iterative method to prove the existence of
$[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}$ code. For $k=3$, we have an
additive $[3,1.5,2]_{4}$ code with generator matrix $A_{[3,1.5,2]_{4}}$ as
follows.

$A_{[3,1.5,2]_{4}}=\begin{pmatrix}w&w&w\\\ 1&1&0\\\ 0&1&1\end{pmatrix}.$ (26)

This is a two-weight code with weight distribution $1+3z^{2}+4z^{3}$, which
has an invariant subcode with parameters $[3,0.5,3]_{4}$.

By Lemma 5, we can get a two-weight additive $[2^{k},\frac{k}{2}+1,3\cdot
2^{k-2}]_{4}$ code, which has a $[2^{k},1,2^{k}]_{4}$ subcode. Let $k=5$, it
is $[8,2.5,6]_{4}$ code, and has an $[8,1,8]_{4}$ subcode. Choosing
$[3,1.5,2]_{2}$ as an auxiliary code, then by Lemma 2, we can get an
$[11,2.5,8]_{4}$. According to Lemma 5, a generator matrix of $[8,2.5,6]_{4}$
code can be denoted as
$A_{[8,2.5,6]_{4}}=\begin{pmatrix}\bm{\gamma_{1}}^{T},\bm{\gamma_{2}}^{T},\bm{\gamma_{3}}^{T},\bm{1_{1,2^{k}-1}}^{T},\bm{w_{1,2^{k}-1}}^{T}\end{pmatrix}^{T}$,
where $\bm{\gamma_{i}}$ are row vectors, $1\leq i\leq 3$. Thus, the generator
matrix of $[11,2.5,8]_{4}$ can be written as

$A_{[11,2.5,8]_{4}}=\left(\begin{array}[]{c|c}\bm{\gamma_{1}}&w,w,w\\\
\bm{\gamma_{2}}&1,1,0\\\ \bm{\gamma_{3}}&0,1,1\\\
\bm{1_{1,2^{k}-1}}&\bm{0_{1,3}}\\\ \bm{w_{1,2^{k}-1}}&\bm{0_{1,3}}\\\
\end{array}\right)=\left(\begin{array}[]{c}\bm{\beta_{1}}\\\ \bm{\beta_{2}}\\\
\bm{\beta_{3}}\\\ \bm{\beta_{4}}\\\ \bm{\beta_{5}}\\\ \end{array}\right).$
(27)

It is easy to judge that $(\bm{\beta_{1}})$ and
$(\bm{\beta_{2}},\bm{\beta_{3}},\bm{\beta_{4}},\bm{\beta_{5}})$ can generate
two additive codes $C_{1}$ and $C_{2}$ with parameters $[11,0.5,9]_{4}$ and
$[11,2,8]_{4}$, respectively, where $C_{[11,2,8]_{4}}$ has weight distribution
$1+15z^{8}$, and $C_{[11,0.5,9]_{4}}$ is an invariant subcode of
$C_{[11,2.5,8]_{4}}$. It follows that $[11,2.5,8]_{4}$ is a two-weight code,
and the weight distribution is $1+15z^{8}+16z^{9}$.

Thus, for $k\geq 7$, by iterative construction we can obtain all codes with
parameters $[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}$ with weight
distribution $1+(2^{k-1}-1)z^{2^{k-2}}+2^{k-1}z^{2^{k-2}+1}$, which has an
invariant subcode with parameters $[\frac{2^{k}+1}{3},0.5,2^{k-2}+1]_{4}$. In
addition, since $g(k,2^{k-1}+2)=2^{k}+k>3(\frac{2^{k}+1}{3}+1)=2^{k}+4$, when
$k\geq 5$, $[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}$ code is GPO. ∎

###### Remark 2

Recently, Bierbrauer et al. [37] also obtained this class of additive codes,
and Lemma 10 can be regarded as a different construction method and rediscover
for the same class of codes.

###### Lemma 11

Let $k\geq 7$ and $k-3\geq k_{2}\geq 4$ be respective odd and even integers.
Then, we have the following results.

(1) There exists a four-weight additive GDO code with parameters
$[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2},2^{k-2}-2^{k_{2}-2}]_{4}$, whose
weight distribution is $1+(2^{k-1}-2^{k_{1}-1})\cdot
z^{2^{k-2}-2^{k_{2}-2}}+(2^{k-1}-2^{k_{1}-1})\cdot
z^{2^{k-2}-2^{k_{2}-2}+1}+(2^{k_{1}-1}-1)z^{k-2}+2^{k_{1}-1}z^{2^{k-2}+1}$,
and has an invariant subcode with parameters
$[\frac{2^{k}-2^{k_{2}}+2}{3},0.5,2^{k-2}-2^{k_{2}-2}+1]_{4}$;

(2) If $k_{2}=4$,
$[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2},2^{k-2}-2^{k_{2}-2}]_{4}$ code is
GDO; otherwise, it is GPO.

###### Proof:

By Theorems 1 and 2, one can construct an additive
$[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k}{2},2^{k-2}-2^{k_{1}-2}-2^{k_{2}-2}]_{4}$
code has two disjoint subcodes
$[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k_{1}}{2},2^{k-2}-2^{k_{1}-2}]_{4}$
and
$[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k_{2}}{2},2^{k-2}-2^{k_{2}-2}]_{4}$.

In addition, Lemma 10 reveals the existence of two-weight
$[\frac{2^{k_{1}}+1}{3},\frac{k_{1}}{2},2^{{k_{1}}-2}]_{4}$ code, which has an
invariant subcode with parameters
$[\frac{2^{k_{1}}+1}{3},0.5,2^{{k_{1}}-2}+1]_{4}$. By virtue of Lemma 2, we
can obtain an additive code with parameters
$[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2},2^{k-2}-2^{k_{2}-2}]_{4}$, whose
generator matrix has the following form

$\begin{array}[]{rl}A_{[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2}]_{4}}=&\begin{pmatrix}A_{[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k_{1}}{2}]_{4}}&A_{[\frac{2^{k_{1}}+1}{3},\frac{k_{1}}{2}]_{4}}\\\
A_{[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k_{2}}{2}]_{4}}&\bm{0_{k_{2}\times(\frac{2^{k_{1}}+1}{3})}}\end{pmatrix}\\\
=&\begin{pmatrix}A_{[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k_{1}}{2}]_{4}}&\begin{matrix}A_{[\frac{2^{k_{1}}+1}{3},0.5]_{4}}\\\
A_{[\frac{2^{k_{1}}+1}{3},\frac{k_{1}}{2}-0.5]_{4}}\end{matrix}\\\
A_{[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k_{2}}{2}]_{4}}&\bm{0_{k_{2}\times(\frac{2^{k_{1}}+1}{3})}}\end{pmatrix},\end{array}$
(28)

where $A_{[\frac{2^{k_{1}}+1}{3},0.5]_{4}}$ is the generator matrix of
invariant subcode
$[\frac{2^{k}-2^{k_{2}}+2}{3},0.5,2^{k-2}-2^{k_{2}-2}+1]_{4}$.

Since the $[\frac{2^{k_{1}}+1}{3},\frac{k_{1}}{2},2^{{k_{1}}-2}]_{4}$ code is
a two-weight code and has an invariant subcode with parameters
$[\frac{2^{k_{1}}+1}{3},0.5,2^{{k_{1}}-2}+1]_{4}$,
$A_{[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2}]_{4}}$ can be divided into two
parts, the first row is an invariant subcode with parameters
$[\frac{2^{k}-2^{k_{2}}+2}{3},0.5]_{4}$, the lower $k-1$ rows are a two-weight
code with weight distribution $1+(2^{k-1}-2^{k_{1}-1})\cdot
z^{2^{k-2}-2^{k_{2}-2}}+(2^{k_{1}-1}-1)\cdot z^{2^{k-2}}$. Therefore,
$A_{[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2}]_{4}}$ generates a four-weight
code with weight distribution $1+(2^{k-1}-2^{k_{1}-1})\cdot
z^{2^{k-2}-2^{k_{2}-2}}+(2^{k-1}-2^{k_{1}-1})\cdot
z^{2^{k-2}-2^{k_{2}-2}+1}+(2^{k_{1}-1}-1)z^{k-2}+2^{k_{1}-1}z^{2^{k-2}+1}$.

In addition, since $g(k,2^{k-1}-2^{k_{2}-1}+2)=2^{k}-2^{k_{2}}+k_{2}+1$ and
$k_{2}\geq 4$, if $k_{2}=4$,
$[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2},2^{k-2}-2^{k_{2}-2}]_{4}$ code is
GDO; otherwise,
$[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2},2^{k-2}-2^{k_{2}-2}]_{4}$ code is
GPO. ∎

###### Lemma 12

Let $k\geq 5$ be an odd number. Suppose that
$A_{[\frac{2^{k}+1}{3},\frac{k}{2}]_{4}}$ and $A_{\frac{k-1}{2}}$ are
separately generator matrices of the code in Lemma 10 and
$\frac{k-1}{2}$-dimensional ASEP code. Then, we have the following results.

(1) The matrix
$\left(A_{[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}}\begin{array}[]{|cc}\bm{1_{2^{k-1}-1}}\\\
A_{\frac{k-1}{2}}\\\ \end{array}\right)$ generates a two-weight additive code
with parameters $[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2},5\cdot 2^{k-3}]_{4}$
and weight distribution $1+(2^{k}-2)z^{5\cdot 2^{k-3}}+z^{3\cdot 2^{k-2}}$.

(2) The additive $[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2},5\cdot
2^{k-3}]_{4}$ code is both GPO and Griesmer code.

###### Proof:

Based on the process of proving Lemmas 5 and 10, the left and right sides of
the above matrix generate additive codes with parameters
$[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}$ and
$[2^{k-1}-1,\frac{k}{2},3\cdot 2^{k-3}-1]_{4}$, respectively, both of which
have two disjoint subcodes. $[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}$ code
has disjoint subcodes $[\frac{2^{k}+1}{3},0.5,2^{k-2}+1]_{4}$ and
$[\frac{2^{k}+1}{3},\frac{k}{2}-0.5,2^{k-2}]_{4}$;
$[2^{k-1}-1,\frac{k}{2},3\cdot 2^{k-3}-1]_{4}$ code has disjoint subcodes
$[2^{k-1}-1,0.5,2^{k-1}-1]_{4}$ and $[2^{k-1}-1,\frac{k}{2}-0.5,3\cdot
2^{k-3}]_{4}$, where the $[2^{k-1}-1,\frac{k}{2}-0.5,3\cdot 2^{k-3}]_{4}$ and
$[\frac{2^{k}+1}{3},\frac{k}{2}-0.5,2^{k-2}]_{4}$ codes are both constant-
weight codes.

Moreover, for any nonzero codewords $\bm{c_{1}}$ of
$C_{[2^{k-1}-1,0.5,2^{k-1}-1]_{4}}$ and $\bm{c_{2}}$ of
$C_{[2^{k-1}-1,\frac{k}{2}-0.5,3\cdot 2^{k-3}]_{4}}$, we have
$\mathbf{w}(\bm{c_{1}}+\bm{c_{2}})=3\cdot 2^{k-3}-1$. For for any nonzero
codewords $\bm{c_{1}^{\prime}}$ of $C_{[\frac{2^{k}+1}{3},0.5,2^{k-2}+1]_{4}}$
and $\bm{c_{2}^{\prime}}$ of
$C_{[\frac{2^{k}+1}{3},\frac{k}{2}-0.5,2^{k-2}]_{4}}$, we get that
$\mathbf{w}(\bm{c_{1}^{\prime}}+\bm{c_{2}^{\prime}})=3\cdot 2^{k-2}+1$.

Therefore, the new code
$[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2},2^{k-2}+3\cdot 2^{k-3}]_{4}$ also
has two disjoint subcodes
$[\frac{2^{k}+1}{3}+2^{k-1}-1,0.5,2^{k-2}+2^{k-1}]_{4}$ and
$[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2}-0.5,2^{k-2}+3\cdot 2^{k-3}]_{4}$,
where $[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2}-0.5,2^{k-2}+3\cdot
2^{k-3}]_{4}$ code has constant-weight.

Furthermore, for any nonzero codewords $\bm{c_{1}^{\prime\prime}}$ of
$C_{[\frac{2^{k}+1}{3}+2^{k-1}-1,0.5]_{4}}$ and $\bm{c_{2}^{\prime\prime}}$ of
$C_{[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2}-0.5]_{4}}$, we can deduce that
$\mathbf{w}(\bm{c_{1}^{\prime\prime}}+\bm{c_{2}^{\prime\prime}})=2^{k-2}+3\cdot
2^{k-3}$. Hence, the $[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2},2^{k-2}+3\cdot
2^{k-3}]_{4}$ code has weight distribution $1+(2^{k}-2)z^{5\cdot
2^{k-3}}+z^{3\cdot 2^{k-2}}$.

In addition, since $g(k,5\cdot 2^{k-2})=5\cdot
2^{k-1}-2=3(\frac{2^{k}+1}{3}+2^{k-1}-1)$, and $g(k,5\cdot 2^{k-2}+2)=5\cdot
2^{k-1}+k-2>3(\frac{2^{k}+1}{3}+2^{k-1})$ for $k\geq 5$,
$[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2},5\cdot 2^{k-3}]_{4}$ code is an
additive both GPO and Griesmer code.

∎

###### Lemma 13

Let $k\geq 7$ and $k_{2}\geq 4$ be odd and even numbers, respectively. Suppose
$A_{[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2}]_{4}}$ and $A_{\frac{k-1}{2}}$
are generator matrices of the code in Lemma 11 and $\frac{k-1}{2}$-dimensional
ASEP code, respectively. Then, we have the following results.

(1) The matrix
$\left(A_{[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2}]_{4}}\begin{array}[]{|cc}\bm{1_{2^{k-1}-1}}\\\
A_{\frac{k-1}{2}}\\\ \end{array}\right)$ generates a two-weight additive code
with parameters $[\frac{2^{k}-2^{k_{2}}+2}{3}+2^{k-1}-1,\frac{k}{2},5\cdot
2^{k-3}-2^{k_{2}-2}]_{4}$ with weight distribution $1+(2^{k}-2)z^{5\cdot
2^{k-3}-2^{k_{2}-2}}+z^{3\cdot 2^{k-2}-2^{k_{2}-2}}$.

(2) The additive $[\frac{2^{k}-2^{k_{2}}+2}{3}+2^{k-1}-1,\frac{k}{2},5\cdot
2^{k-3}-2^{k_{2}-2}]_{4}$ code is both GPO and Griesmer code.

###### Proof:

Similarly, this lemma can be proved using the same method as Lemma 12. ∎

So far, we construct ten classes of optimal additive codes and determine their
weight distribution. For the reader’s convenience, we summarize additive GPO
and Griesmer additive codes in Table I. In particular, we omit some relatively
trivial parameters that are not listed, such as $[2^{k}-1,\frac{k}{2}+1,3\cdot
2^{k-2}-1]_{4}$ code in Lemma 5 that can be directly derived from No. 2 in
Table I. These codes are superior to linear codes in two ways, on the one hand
to optimal linear codes [38, 39] and on the other hand to the linear few-
weight codes [27, 32].

TABLE I: Quaternary additive Griesmer puncture-optimal codes with few weights
and $k=k_{1}+k_{2}$

No. | Additive Codes | Weight Distribution | Constructions | Ranges
---|---|---|---|---
1 | $\bm{[2^{k}-1,\frac{k}{2},3\cdot 2^{k-2}]_{4}}^{\diamond}$ | $\bm{1+2^{k}\cdot z^{3\cdot 2^{k-2}}}$ |  Lemma 3 | $k\geq 3$
2 | $\bm{[2^{k},\frac{k}{2}+1,3\cdot 2^{k-2}]_{4}}^{\diamond}$ | $\bm{1+(2^{k+2}-4)\cdot z^{3\cdot 2^{k-2}}+3z^{2^{k}}}$ |  Lemma 5 | $k\geq 3$
3 | $\bm{[2^{k}-1-\frac{2^{k_{2}}-1}{3},\frac{k}{2},3\cdot 2^{k-2}-2^{k_{2}-2}]_{4}}^{\diamond}$ |  $\bm{1+(2^{k}-2^{k_{1}})\cdot z^{3\cdot 2^{k-2}-2^{k_{2}-2}}}$ $\bm{+(2^{k_{1}}-1)\cdot z^{3\cdot 2^{k-2}}}$ |  Corollary 1 |  $k\geq 7$ $k_{2}\geq 4$
4 | $\bm{[2^{k}-1-m\cdot\frac{2^{k_{2}}-1}{3},\frac{k}{2},3\cdot 2^{k-2}-m\cdot 2^{k_{2}-2}]_{4}}$ |  $\bm{1+(2^{k}-2^{k_{1}})\cdot z^{3\cdot 2^{k-2}-m\cdot 2^{k_{2}-2}}}$ $\bm{+(2^{k_{1}}-1)\cdot z^{3\cdot 2^{k-2}}}$ |  Corollary 1 |  $\bm{k\geq 7}$ $\bm{k_{2}\geq 4}$ $\bm{2\leq m\leq 3}$
5 | $\bm{[\frac{2^{k}-2^{k_{1}}}{3},\frac{k}{2},2^{k-2}-2^{k_{1}-2}]_{4}}^{\diamond}$ | $\bm{1+(2^{k}-2^{k_{2}})\cdot z^{2^{k-2}-2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{2^{k-2}}}$ |  Theorem 2 |  $\bm{k_{1}\geq 3}$ $\bm{k_{2}\geq 4}$
6 | $\bm{[\frac{2^{k}-2^{k_{1}}-2^{k_{2}}+1}{3},\frac{k}{2},2^{k-2}-2^{k_{1}-2}-2^{k_{2}-2}]_{4}}^{\diamond}$ |  $1+(2^{k}-2^{k_{1}}-2^{k_{2}})\cdot z^{2^{k-2}-2^{k_{1}-2}-2^{k_{2}-2}}+$ $(2^{k_{1}}-1)\cdot z^{2^{k-2}-2^{k_{1}-2}}+(2^{k_{2}}-1)\cdot z^{2^{k-2}-2^{k_{2}-2}}$ |  Theorem 2 |  $k_{1}\geq 3$ $k_{2}\geq 4$
7 | $\bm{[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}}$ | $\bm{1+(2^{k-1}-1)z^{2^{k-2}}+2^{k-1}z^{2^{k-2}+1}}$ |  Lemma 10 | $\bm{k\geq 5}$
8 | $\bm{[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2},2^{k-2}-2^{k_{2}-2}]_{4}}$ |  $\bm{1+(2^{k-1}-2^{k_{1}-1})\cdot z^{2^{k-2}-2^{k_{2}-2}}+}$ $\bm{(2^{k-1}-2^{k_{1}-1})\cdot z^{2^{k-2}-2^{k_{2}-2}+1}}$ $\bm{+(2^{k_{1}-1}-1)z^{k-2}+2^{k_{1}-1}z^{2^{k-2}+1}}$ |  Lemma 11 |  $\bm{k\geq k_{2}+3}$ $\bm{k_{2}\geq 6}$
9 | $\bm{[\frac{2^{k}+1}{3}+2^{k-1}-1,\frac{k}{2},5\cdot 2^{k-3}]_{4}}^{\diamond}$ | $\bm{1+(2^{k}-2)z^{5\cdot 2^{k-3}}+z^{3\cdot 2^{k-2}}}$ |  Lemma 12 | $k\geq 5$
10 | $\bm{[\frac{2^{k}-2^{k_{2}}+2}{3}+2^{k-1}-1,\frac{k}{2},5\cdot 2^{k-3}-2^{k_{2}-2}]_{4}}^{\diamond}$ | $\bm{1+(2^{k}-2)z^{5\cdot 2^{k-3}-2^{k_{2}-2}}+z^{3\cdot 2^{k-2}-2^{k_{2}-2}}}$ |  Lemma 13 |  $k\geq 7$

  * Note: According to Lemma 1, additive codes in this table are all additive GPO codes. The superscript “$\diamond$” indicates that the corresponding is also an additive Griesmer code.

###### Remark 3

It should be noted that by the additive codes in Table I and Lemma 7, one can
cover all the results of additive GPO codes with non-integer dimensions in
[16] and [18]. Details are as follows.

  * •

$[8,2.5,6]_{4}$ (No.2, $k=3$)

  * •

$[11,2.5,8]_{4}$ (No.7, $k=5$)

  * •

$[26,2.5,20]_{4}$ (No.9, $k=5$)

  * •

$[31,2.5,24]_{4}$ (No.1, $k=5$)

  * •

$[32,3.5,24]_{4}$ (No.2, $k=5$)

  * •

$[35,3.5,26]_{4}$ (No.6, $k=7$, $k_{1}=3$)

  * •

$[40,3.5,30]_{4}$ (No.5, $k=7$, $k_{1}=3$)

  * •

$[43,3.5,32]_{4}$ (No.7, $k=7$)

  * •

$[127,3.5,96]_{4}$ (No.1, $k=7$)

  * •

$[128,4.5,96]_{4}$ (No.2, $k=7$)

  * •

$[155,4.5,116]_{4}$ (No.6, $k=9$, $k_{1}=5$)

  * •

$[160,4.5,120]_{4}$ (No.5, $k=9$, $k_{1}=5$)

  * •

$[168,4.5,126]_{4}$ (No.5, $k=9$, $k_{1}=3$)

  * •

$[171,4.5,128]_{4}$ (No.7, $k=9$)

  * •

$[163,4.5,122]_{4}$ (using $[35,3.5,26]_{4}$ code by Lemma 7)

Specifically, $[8,2.5,6]_{4}$, $[11,2.5,8]_{4}$, $[26,2.5,20]_{4}$ and
$[31,2.5,24]_{4}$ codes are the quaternary additive GPO codes constructed in
[16], while the others are results in [18].

##  VI Constructions of 3.5-dimensional optimal quaternary additive codes

In this section, we employ the additive codes obtained in the previous
sections and associated construction methods to construct quaternary
3.5-dimensional optimal additive codes.

###### Definition 4

Let $C_{1}$ be a linear optimal $[n,k,d]_{4}^{l}$ code. If there exists a
linear optimal code $C_{2}$ with parameters $[n,k-1,d+1]^{l}_{4}$ satisfying
$C_{2}\subset C_{1}$, then we call $C_{1}$ as a linear $X$-optimal code.

###### Lemma 14

If there exists a quaternary linear $X$-optimal code $C_{x}$ with parameters
$[n_{x},k,d_{x}]_{4}^{l}$, then there also exist additive codes with
parameters $[\frac{2^{2k-1}+1}{3}+n_{x},k-0.5,2^{2k-3}+d_{x}+1]_{4}$ and
$[\frac{2^{2k-1}-2^{k_{2}}+2}{3}+n_{x},k-0.5,2^{2k-3}-2^{k_{2}-2}+d_{x}+1]_{4}$,
where $k_{2}<2k-1$ is an even number.

###### Proof:

Since all linear codes are additive, $C_{x}$ also can be written as an
additive code $[n,k,d]_{4}$. According to Definition 4, $C_{x}$ has an
additive subcode $C_{x}^{\prime}$ with parameters $[n,k-0.5,d]_{4}$, satisfy
$[n,k-1,d+1]_{4}\subset[n,k-0.5,d]_{4}$. From Lemmas 10 and 11, we have two
classes of additive codes with parameters
$[\frac{2^{k}+1}{3},\frac{k}{2},2^{k-2}]_{4}$ and
$[\frac{2^{k}-2^{k_{2}}+2}{3},\frac{k}{2},2^{k-2}-2^{k_{2}-2}]_{4}$, and
invariant subcodes of them have parameters
$[\frac{2^{k}+1}{3},0.5,2^{k-2}+1]_{4}$ and
$[\frac{2^{k}-2^{k_{2}}+2}{3},0.5,2^{k-2}-2^{k_{2}-2}+1]_{4}$, respectively.
By Corollary 2, the combination of $C_{x}$ and these two classes of additive
codes can produce additive codes with parameters
$[\frac{2^{2k-1}+1}{3}+n_{x},k-0.5,2^{2k-3}+d_{x}+1]_{4}$ and
$[\frac{2^{2k-1}-2^{k_{2}}+2}{3}+n_{x},k-0.5,2^{2k-3}-2^{k_{2}-2}+d_{x}+1]_{4}$.
∎

###### Example 2

In [38], there is an $X$-optimal code with parameters $[16,4,11]_{4}^{l}$,
which satisfies $[16,3,12]_{4}^{l}\subset[16,4,11]_{4}^{l}$. Write the
generator matrix of $X$-optimal $[16,4,11]_{4}^{l}$ code in the additive form
(it is an $8\times 16$ matrix) and choose the lower $7$-rows as
$A_{[16,3.5]_{4}}$, such that the lower 6-rows of $A_{[16,3.5]_{4}}$ generates
an additive $[16,3,12]_{4}$ code.

${\scriptsize
A_{[16,3.5]_{4}}=\left({\begin{array}[]{*{16}{c}}0&0&0&1&1&w&0&1&w&w^{2}&w^{2}&w&1&0&w&1\\\
1&0&0&1&0&w^{2}&w&1&w^{2}&1&0&1&w^{2}&1&w&w^{2}\\\
w&0&0&w&0&1&w^{2}&w&1&w&0&w&1&w&w^{2}&1\\\
0&1&0&w^{2}&w&1&w^{2}&1&0&1&w^{2}&1&w&w^{2}&0&1\\\
0&w&0&1&w^{2}&w&1&w&0&w&1&w&w^{2}&1&0&w\\\
0&0&1&1&w&0&1&w&w^{2}&w^{2}&w&1&0&w&1&1\\\
0&0&w&w&w^{2}&0&w&w^{2}&1&1&w^{2}&w&0&w^{2}&w&w\end{array}}\right)},$ (29)

${\tiny
A_{[38,3.5,28]_{4}}=\left({\begin{array}[]{*{43}{c}}1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&w&w&w\\\
w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&1&0&1\\\
0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&w&w^{2}&w&1&1&w^{2}&0&1&1\\\
1&1&1&1&1&1&1&0&0&0&0&0&0&0&1&1&1&1&1&1&1&w&w&w&w&w&w&w&w&w&w&w&w&w&w&0&0&0\\\
w&w&w&w&w&w&w&0&0&0&0&0&0&0&w&w&w&w&w&w&w&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&0&0&0\\\
0&0&0&0&0&0&0&1&1&1&1&1&1&1&w&w&w&w&w&w&w&w&w&w&w&w&w&w&1&1&1&1&1&1&1&0&0&0\\\
0&0&0&0&0&0&0&w&w&w&w&w&w&w&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w^{2}&w&w&w&w&w&w&w&0&0&0\end{array}}\right),}$
(30) ${\tiny
A_{[43,3.5,32]_{4}}=\left({\begin{array}[]{*{43}{c}}0&1&0&0&1&w&w^{2}&w^{2}&0&0&w^{2}&1&1&1&w^{2}&0&w&0&w^{2}&w^{2}&w&w^{2}&1&w^{2}&w&1&w&1&0&w&w&w&0&1&0&w^{2}&w^{2}&1&w&w&w&w&w\\\
0&w&0&1&w&1&w&w^{2}&1&1&w^{2}&w&w^{2}&w^{2}&w&1&0&0&w^{2}&w^{2}&0&w&w&w&0&w^{2}&1&w^{2}&0&0&1&1&0&w&1&w^{2}&w&w^{2}&0&1&1&1&0\\\
0&0&1&0&w&0&1&w&1&w&w^{2}&1&1&w^{2}&w&w^{2}&w^{2}&w&1&0&0&w^{2}&w^{2}&0&w&w&w&0&w^{2}&1&w^{2}&1&0&0&w&1&w^{2}&w&w^{2}&1&1&0&1\\\
0&0&w&1&w^{2}&1&w&w&w^{2}&w&0&w&w^{2}&0&w^{2}&0&1&w^{2}&w^{2}&1&0&1&1&0&w^{2}&w&w&1&1&w&0&w^{2}&1&1&1&1&1&1&1&1&0&0&0\\\
0&0&0&w&1&w^{2}&1&w&w&w^{2}&w&0&w&w^{2}&0&w^{2}&0&1&w^{2}&w^{2}&1&0&1&1&0&w^{2}&w&w&1&1&w&w^{2}&w&w&w&w&w&w&w&w&0&0&0\\\
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0\\\
w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&w&0&0&0&0&0&0&0&0&0&0&0\end{array}}\right).}$
(31)

By Lemma 14, combining $A_{[16,3.5]_{4}}$ with the generator matrices
$A_{[38,3.5,28]_{4}}$ of $[38,3.5,28]_{4}$ code in Lemma 11 and
$A_{[43,3.5,32]_{4}}$ of $[43,3.5,32]_{4}$ code in Lemma 10, we can get
additive GPO codes with parameters $[54,3.5,40]_{4}$ and $[59,3.5,44]_{4}$,
respectively.

In particular, using Magma [40], one can determine that both of them are two-
weight codes whose weight distributions are $1+101z^{40}+26z^{44}$ and
$1+108z^{44}+19z^{48}$, respectively.

Next, let us solve the problem of constructing all optimal quaternary
$3.5$-dimensional additive codes. As mentioned in Sect. I, this is to
construct all GPO or PO codes with dimension $3.5$. Firstly, Lemma 1 plays a
vital role in determining the upper bound on GPO codes. However, the additive
Griesmer bound is typically not tight for codes with short lengths. Here, we
use inverse methods to determine the upper bound on PO codes, as shown in the
following lemma.

###### Lemma 15

If there is no linear $[3n,2k,2d]^{l}_{2}$ code, then there is no additive
$[n,k,d]_{4}$ code.

###### Proof:

If additive $[n,k,d]_{4}$ code exists, then by concatenating with linear
$[3,2,2]_{2}^{l}$ code, we can derive a linear $[3n,2k,2d]^{l}_{2}$ code.
Therefore, if $[3n,2k,2d]^{l}_{2}$ code does not exist, $[n,k,d]_{4}$ does not
exist either. ∎

###### Corollary 3

There are no quaternary additive codes with the following parameters.

(1) For $n=18,19$, additive $[n,7/2,n-5]_{4}$ code;

(2) For $n=26,27$, additive $[n,7/2,n-7]_{4}$ code.

###### Proof:

These codes satisfy Lemma 1, but according to [38], there is no
$[54,7,26]^{l}_{2}$, $[57,7,28]^{l}_{2}$, $[78,7,38]^{l}_{2}$ and
$[81,7,40]^{l}_{2}$ codes. Therefore, the above additive codes violate Lemma
15. ∎

###### Lemma 16

If additive $[n,\frac{k}{2},d]_{4}$ code is GPO, then additive
$[n+2^{k}-1,\frac{k}{2},d+3\cdot 2^{k-2}]_{4}$ code is also GPO.

###### Proof:

Since $[n,\frac{k}{2},d]_{4}$ code is an GPO, we have that additive
$[n+1,\frac{k}{2},d+1]_{4}$ code violates Lemma 1, i.e.,
$3n+3>\sum\limits_{i=0}^{k-1}\left\lceil\frac{d+1}{2^{i-1}}\right\rceil$.
Since $\sum\limits_{i=0}^{k-1}\left\lceil\frac{d+3\cdot
2^{k-2}}{2^{i-1}}\right\rceil=2d+3\cdot 2^{k-1}+d+3\cdot
2^{k-2}+\cdots+\left\lceil\frac{d}{2^{k-2}}\right\rceil+3=\sum\limits_{i=0}^{k-1}\left\lceil\frac{d}{2^{i-1}}\right\rceil+3(2^{k}-1)$,
$[n+2^{k},\frac{k}{2},d+3\cdot 2^{k-2}+1]_{4}$ code also violates Lemma 1,
i.e., $[n+2^{k}-1,\frac{k}{2},d+3\cdot 2^{k-2}]_{4}$ code is also an GPO code.
∎

With the help of Lemma 16, we can transform the problem of determining all
quaternary $\frac{k}{2}$-dimensional long GPO codes into the problem of
constructing all GPO codes of lengths between $i(2^{k}-1)$ and
$(i+1)(2^{k}-1)$, where $i$ is a non-negative integer. Here, we call $2^{k}-1$
as the period of quaternary $\frac{k}{2}$-dimensional additive codes.

By corollary 3, there are no $3.5$-dimensional GPO codes of lengths $19$ and
$27$. Thus, to solve the construction problem of all 3.5-dimensional optimal
additive codes, we need to construct at least all GPO or PO codes in the first
two periods, i.e., determine all 3.5-dimensional GPO and PO codes with lengths
up to $254$. In Table II, we list them of lengths between $4$ and $254$,
except for three undetermined ones.

In Table II, we use $C_{t}$ denote additive $[n,3.5,n-t]_{4}$ codes with
variable $n$, and $(C_{t_{1}}\mid C_{t_{2}})$ denote the juxtaposition code of
$C_{t_{1}}$ and $C_{t_{2}}$, in which $C_{t_{1}}$ and $C_{t_{2}}$ are
defaulted to the maximum lengths. $C_{t}$ with the largest $n$ is GPO or PO
code, others are GDO or DO codes. In Table II, if $t=5$, $6$ or $\geq 8$, then
the code $C_{t}$ with largest $n$ are all GPO and the others are PO. Under
Lemma 16, for $n\geq 255$, we can construct all GPO codes, except for
$t=6,7,12$, by combining the codes in Table II with $[127,3.5,96]_{4}$ code in
Lemma 3, and all GDO codes can be obtained by puncturing them.

TABLE II: Optimal quaternary additive 3.5-dimensional codes of lengths from
$4$ to $254$

No. | $C_{t}$ | parameters | Range | Constructions
---|---|---|---|---
1 | $C_{3}$ | $[n,3.5,n-3]_{4}$ | $4\leq n\leq 7$ | [10]
2 | $C_{4}$ | $[n,3.5,n-4]_{4}$ | $8\leq n\leq 12$ | [10]
3 | $C_{5}$ | $[n,3.5,n-5]_{4}$ | $13\leq n\leq 17$ | [38]
4 | $C_{6}$ | $[n,3.5,n-6]_{4}$ | $18\leq n\leq 22$ | -
5 | $C_{7}$ | $[n,3.5,n-7]_{4}$ | $23\leq n\leq 25$ | -
6 | $C_{8}$ | $[n,3.5,n-8]_{4}$ | $26\leq n\leq 32$ | Lemma 5
7 | $C_{9}$ | $[n,3.5,n-9]_{4}$ | $33\leq n\leq 35$ | Theorem 2
8 | $C_{10}$ | $[n,3.5,n-10]_{4}$ | $36\leq n\leq 40$ | Theorem 2
9 | $C_{11}$ | $[n,3.5,n-11]_{4}$ | $41\leq n\leq 43$ | Lemma 10
10 | $C_{12}$ | $[n,3.5,n-12]_{4}$ | $44\leq n\leq 46$ | -
11 | $C_{13}$ | $[n,3.5,n-13]_{4}$ | $47\leq n\leq 51$ | [19]
12 | $\bm{C_{14}}$ | $\bm{[n,3.5,n-14]_{4}}$ | $\bm{52\leq n\leq 54}$ |  Example 2
13 | $\bm{C_{15}}$ | $\bm{[n,3.5,n-15]_{4}}$ | $\bm{55\leq n\leq 59}$ |  Example 2
14 | $C_{16}$ | $[n,3.5,n-16]_{4}$ | $60\leq n\leq 64$ | Lemma 5
15 | $C_{17}$ | $[n,3.5,n-17]_{4}$ | $65\leq n\leq 67$ | $(C_{8}\mid C_{9})$
16 | $C_{18}$ | $[n,3.5,n-18]_{4}$ | $68\leq n\leq 72$ | $(C_{8}\mid C_{10})$
17 | $C_{19}$ | $[n,3.5,n-19]_{4}$ | $73\leq n\leq 75$ | $(C_{8}\mid C_{11})$
18 | $C_{20}$ | $[n,3.5,n-20]_{4}$ | $76\leq n\leq 80$ | $(C_{10}\mid C_{10})$
19 | $C_{21}$ | $[n,3.5,n-21]_{4}$ | $81\leq n\leq 85$ | Lemma 7
20 | $C_{22}$ | $[n,3.5,n-22]_{4}$ | $86\leq n\leq 86$ | Extend $C_{21}$
21 | $\bm{C_{23}}$ | $\bm{[n,3.5,n-23]_{4}}$ | $\bm{87\leq n\leq 91}$ | $\bm{(C_{8}\mid C_{15})}$
22 | $C_{24}$ | $[n,3.5,n-24]_{4}$ | $92\leq n\leq 96$ | Lemma 5
23 | $\bm{C_{25}}$ | $\bm{[n,3.5,n-25]_{4}}$ | $\bm{97\leq n\leq 101}$ |  Lemma 13
24 | $\bm{C_{26}}$ | $\bm{[n,3.5,n-26]_{4}}$ | $\bm{102\leq n\leq 106}$ |  Lemma 12
25 | $C_{27}$ | $[n,3.5,n-27]_{4}$ | $107\leq n\leq 107$ | Extend $C_{26}$
26 | $C_{28}$ | $[n,3.5,n-28]_{4}$ | $108\leq n\leq 112$ | Corollary 1
27 | $\bm{C_{29}}$ | $\bm{[n,3.5,n-29]_{4}}$ | $\bm{113\leq n\leq 117}$ |  Corollary 1
28 | $\bm{C_{30}}$ | $\bm{[n,3.5,n-30]_{4}}$ | $\bm{118\leq n\leq 122}$ |  Corollary 1
29 | $C_{31}$ | $[n,3.5,n-31]_{4}$ | $123\leq n\leq 127$ | Lemma 3
30 | $C_{32}$ | $[n,3.5,n-32]_{4}$ | $128\leq n\leq 128$ | Extend $C_{31}$
31 | $\bm{C_{33}}$ | $\bm{[n,3.5,n-33]_{4}}$ | $\bm{129\leq n\leq 133}$ | $\bm{(C_{8}\mid C_{25})}$
32 | $\bm{C_{34}}$ | $\bm{[n,3.5,n-34]_{4}}$ | $\bm{134\leq n\leq 138}$ | $\bm{(C_{8}\mid C_{26})}$
33 | $\bm{C_{35}}$ | $\bm{[n,3.5,n-35]_{4}}$ | $\bm{139\leq n\leq 141}$ | $\bm{(C_{10}\mid C_{25})}$
34 | $\bm{C_{36}}$ | $\bm{[n,3.5,n-36]_{4}}$ | $\bm{142\leq n\leq 146}$ | $\bm{(C_{10}\mid C_{26})}$
35 | $\bm{C_{37}}$ | $\bm{[n,3.5,n-37]_{4}}$ | $\bm{147\leq n\leq 149}$ | $\bm{(C_{11}\mid C_{26})}$
36 | $\bm{C_{38}}$ | $\bm{[n,3.5,n-38]_{4}}$ | $\bm{150\leq n\leq 154}$ | $\bm{(C_{8}\mid C_{30})}$
37 | $C_{39}$ | $[n,3.5,n-39]_{4}$ | $155\leq n\leq 159$ | $(C_{8}\mid C_{31})$
38 | $C_{40}$ | $[n,3.5,n-40]_{4}$ | $160\leq n\leq 162$ | $(C_{9}\mid C_{31})$
39 | $C_{41}$ | $[n,3.5,n-41]_{4}$ | $163\leq n\leq 167$ | $(C_{10}\mid C_{31})$
40 | $C_{42}$ | $[n,3.5,n-42]_{4}$ | $168\leq n\leq 170$ | $(C_{11}\mid C_{31})$
41 | $\bm{C_{43}}$ | $\bm{[n,3.5,n-43]_{4}}$ | $\bm{171\leq n\leq 173}$ | $\bm{(C_{8}\mid C_{35})}$
42 | $\bm{C_{44}}$ | $\bm{[n,3.5,n-44]_{4}}$ | $\bm{174\leq n\leq 178}$ | $\bm{(C_{8}\mid C_{36})}$
43 | $\bm{C_{45}}$ | $\bm{[n,3.5,n-45]_{4}}$ | $\bm{179\leq n\leq 181}$ | $\bm{(C_{14}\mid C_{31})}$
44 | $\bm{C_{46}}$ | $\bm{[n,3.5,n-46]_{4}}$ | $\bm{182\leq n\leq 186}$ | $\bm{(C_{15}\mid C_{31})}$
45 | $C_{47}$ | $[n,3.5,n-47]_{4}$ | $187\leq n\leq 191$ | $(C_{16}\mid C_{31})$
46 | $C_{48}$ | $[n,3.5,n-48]_{4}$ | $192\leq n\leq 194$ | $(C_{17}\mid C_{31})$
47 | $C_{49}$ | $[n,3.5,n-49]_{4}$ | $195\leq n\leq 199$ | $(C_{18}\mid C_{31})$
48 | $C_{50}$ | $[n,3.5,n-50]_{4}$ | $200\leq n\leq 202$ | $(C_{19}\mid C_{31})$
49 | $C_{51}$ | $[n,3.5,n-51]_{4}$ | $203\leq n\leq 207$ | $(C_{20}\mid C_{31})$
50 | $C_{52}$ | $[n,3.5,n-52]_{4}$ | $208\leq n\leq 212$ | $(C_{21}\mid C_{31})$
51 | $C_{53}$ | $[n,3.5,n-53]_{4}$ | $213\leq n\leq 213$ | $(C_{22}\mid C_{31})$
52 | $\bm{C_{54}}$ | $\bm{[n,3.5,n-54]_{4}}$ | $\bm{214\leq n\leq 218}$ | $\bm{(C_{23}\mid C_{31})}$
53 | $C_{55}$ | $[n,3.5,n-55]_{4}$ | $219\leq n\leq 223$ | $(C_{24}\mid C_{31})$
54 | $\bm{C_{56}}$ | $\bm{[n,3.5,n-56]_{4}}$ | $\bm{224\leq n\leq 228}$ | $\bm{(C_{25}\mid C_{31})}$
55 | $\bm{C_{57}}$ | $\bm{[n,3.5,n-57]_{4}}$ | $\bm{229\leq n\leq 233}$ | $\bm{(C_{26}\mid C_{31})}$
58 | $C_{60}$ | $[n,3.5,n-58]_{4}$ | $234\leq n\leq 234$ | $(C_{27}\mid C_{31})$
59 | $C_{61}$ | $[n,3.5,n-59]_{4}$ | $235\leq n\leq 239$ | $(C_{28}\mid C_{31})$
60 | $\bm{C_{62}}$ | $\bm{[n,3.5,n-60]_{4}}$ | $\bm{240\leq n\leq 244}$ | $\bm{(C_{29}\mid C_{31})}$
61 | $\bm{C_{63}}$ | $\bm{[n,3.5,n-61]_{4}}$ | $\bm{245\leq n\leq 249}$ | $\bm{(C_{30}\mid C_{31})}$
62 | $C_{64}$ | $[n,3.5,n-62]_{4}$ | $250\leq n\leq 254$ | $(C_{31}\mid C_{31})$

  * Note: The marked bold indicates the new quaternary additive codes obtained in this paper.

##  VII Conclusion

In this work, we studied combinatorial techniques for quaternary additive
codes and successfully derived ten classes of optimal additive codes with non-
integer dimensions, see Table I. Specifically, we detailed the combinatorial
construction of additive constant-weight codes and elucidated the algebraic
structure of their generator matrices, see Lemmas 3, 6 and 9. Then, we
proposed an anticode construction method for additive codes, see Lemma 8 and
Theorem 1. Furthermore, we present a generalized Construction X that enables
the construction of additive codes with non-integer dimensions using
specialized optimal linear codes, see Lemma 14 and Theorem 3. Finally, we also
determined the optimal additive $[n,3.5,n-t]_{4}$ codes for all $t$ with
variable $n$, except for $t=6,7,12$, see Lemma 16 and Table II.

## Acknowledgments

The authors would like to thank Dr. Shitao Li and Dr. Yang Li for the helpful
discussions.

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