Optimal quaternary Hermitian self-orthogonal $[n, 5]$ codes of $n \geq 492$

1.   Introduction

Let $F^{n}_{q}$ be the $n$-dimensional vector space over the Galois field $F_{q}$ $= GF(q)$. An $[n, k, d]_{q}$ code is a $k$-dimensional subspace of $F^{n}_{q}$ with minimal distance $d$. An $[n, k, d]_{q}$ code is optimal if there is no $[n, k, d+1]_{q}$ code. Parameters of optimal $[n, k]_{q}$ codes for the following $k$ and $q$ are solved ^[1,2,3,4]:

1) $q = 2$ and $k\leq 8$;

2) $q = 3$ and $k\leq 5$;

3) $q = 4$ and $k\leq 4$, there are 104 open cases for $k = 5$ and $n \leq 492$.

Self-orthogonal (SO) codes, including self-dual codes, form an important class of linear codes. Such codes have close connections to other mathematical structures such as block designs, lattices, and sphere packings ^[5,6,7], which can also be used to construct quantum codes, see ^[8,9] and references therein. An $[n, k, d]_{q}$ SO code is optimal if there is no $[n, k, d_{so}]_{q}$ SO code with $d_{so} > d$. So, people try to solve the problem of determining low-dimensional optimal SO codes over small fields as it is done for optimal linear codes in ^[1,2,3,4].

There are some achievements on low-dimensional optimal SO codes over $F_{2}$–$F_{4}$. Pless classified certain binary optimal SO codes for $n\leq 20$ in ^[10]. Bouyukliev et al. determined binary optimal $[n, k]_{2}$ SO codes for $k\leq 3$, and classified some optimal SO codes with $n\leq 40$ in ^[11]. Binary optimal $[n, k]_{2}$ SO codes were solved in ^[12,13] for $k = 4$, and in ^[14,15,16] for $k = 5, 6$. Shi et al. also determined parameters of most binary optimal SO codes with dimension $k = 7, 8$ in ^[16]. Bouyukliev et al. classified certain ternary and quaternary optimal SO codes with $n\leq 29$ and $k\leq 6$ in ^[17]. Guan et al. constructed some ternary optimal SO codes from known self-dual codes in ^[18]. Li et al. determined parameters of ternary optimal SO codes with dimension $k\leq 5$, except two special $[n, 5]_{3}$ SO codes ^[19]. Following ^[17], Ma et al. established $[n, k]_{4}$ quaternary optimal HSO codes for $k\leq 3$ in ^[20] by constructing such codes. In ^[21], Ren et al. gave exact parameters for $[n, 4]_{4}$ optimal HSO codes for all but two values of $n$. For most $n$ with $n\leq 492$, $[n, 5]_{4}$ optimal HSO codes are determined, and there are also 108 cases remaining open, see^[22].

A well-known lower bound on $[n, k, d]_{q}$ optimal linear codes, is called the Griesmer bound as follows:

According to ^[2,3], for $q = 4$, and $k = 5$, the Griesmer bound is achieved when $d\geq 369$, hence all optimal $[n, 5]_{4}$ codes are known for $n \geq 492$. In this paper, we consider optimal $[n, 5]_{4}$ HSO codes for $n \geq 492$, and our result is the following theorem.

Theorem 1. Let $n \geq 492$. If there is an $[n, 5, d_{0}(n, 5)]_{4}$ optimal linear code, then there is an $[n, 5, 2 \lfloor d_{0}(n, 5)/2\rfloor ]_{4}$ optimal HSO code.

2.   Preliminaries

In this section, we prepare some notations and basic results used in this paper.

Let $F_{4} = \left\{0, 1, \omega, \omega^2\right\}$ be the field with four elements, where $\omega^2 = 1+\omega$. For $x$$\in F_{4}$, its conjugate is $\overline{x} = x^{2}$. For $u = (u_{1}, u_{2}, \cdots, u_{n})$, $v = (v_{1}, v_{2}, \cdots, v_{n})\in F_{4}^{n}$, their Hermitian inner product is $(u, v)_{h} = \sum_{i = 1}^{n}u_{i}\cdot \overline{v_{i}} = \sum_{i = 1}^{n}u_{i}\cdot v_{i}^{2}$. The Hermitian dual code $\mathcal{C}$$^{\perp_{h}}$ of $\mathcal{C}$ is defined as $\mathcal{C}^{\perp_{h}} = \{u\in F_{4}^{n}\mid (u, v)_{h} = 0, \forall v\in \mathcal{C}\}$. If $\mathcal{C} \subseteq \mathcal{C}^{\perp_{h}}$, then $\mathcal{C}$ is called an HSO code. Especially, if $\mathcal{C} = \mathcal{C}^{\perp_{h}}$, then $\mathcal{C}$ is a Hermitian self-dual code.

For given $k$, if $n\geq 2k$, there is an $[n, k]$ HSO code over $F_{4}$, which is an even code. For the sake of simplicity, we use 2 and 3 to represent $\omega$ and $\omega^2$ in the rest of this paper, respectively. Let $\bf{1_{n}}$ = $(1, 1, \ldots, 1)_{1\times n}$ and $\bf{0_{n}}$ = $(0, 0, \ldots, 0)_{1\times n}$ denote the all-one vector and the all-zero vector of length $n$, respectively.

For an $m\times n$ matrix $M$, define $\underline{M}$, $\overline{M}$, $\underline{\underline{M}}$, and $\overline{\overline{M}}$ as the following matrices:

respectively.

Construct

$S_{2} = \left(\begin{array}{cccccc} 10111\\ 01231\\ \end{array} \right),$ $S_{3} = \left(\begin{array}{cccccc} S_{2} & \mathbf{0}_{2\times1}&S_{2}&S_{2}&S_{2}\\ \mathbf{0}_{5} & 1 & \mathbf{1}_{5} & 2\cdot\mathbf{1}_{5} & 3\cdot\mathbf{1}_{5}\\ \end{array} \right),$

$S_{4} = \left(\begin{array}{cccccc} S_{3} & \mathbf{0}_{3\times1}&S_{3}&S_{3}&S_{3}\\ \mathbf{0}_{21} & 1 & \mathbf{1}_{21} & 2\cdot\mathbf{1}_{21} & 3\cdot\mathbf{1}_{21}\\ \end{array} \right) = \left(\begin{array}{cccccc} \underline{ S_{3}}\mid &A_{4}\\ \end{array} \right)$,

$S_{5} = \left(\begin{array}{cccccc} S_{4} & \mathbf{0}_{4\times1}&S_{4}&S_{4}&S_{4}\\ \mathbf{0}_{85} & 1 & \mathbf{1}_{85} & 2\cdot\mathbf{1}_{85} & 3\cdot\mathbf{1}_{85}\\ \end{array} \right) = \left(\begin{array}{cccccc} \underline{ S_{4}}\mid&A_{5}\\ \end{array} \right)$,

$S'_{4} = \left(\begin{array}{cccccc} \mathbf{0}_{21} & 1 & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21}\\ S_{3} & \mathbf{0}_{3\times1}&S_{3} & 2\cdot S_{3} & 3\cdot S_{3}\\ < /p > < p > \end{array} \right) = \left(\begin{array}{cccccc} \overline{S_{3}}\mid&B_{4}\\ \end{array} \right)$,

$S'_{5} = \left(\begin{array}{cccccc} \mathbf{0}_{85} & 1 & \mathbf{1}_{85} & \mathbf{1}_{85} & \mathbf{1}_{85}\\ S_{4} & \mathbf{0}_{4\times1}&S_{4} & 2\cdot S_{4} & 3\cdot S_{4}\\ < /p > < p > \end{array} \right) = \left(\begin{array}{cccccc} \overline{S'_{4}}\mid &B_{5}\\ \end{array} \right) = \left(\begin{array}{cccccc} \overline{\overline{S_{3}}}\mid & \overline{B_{4}}\mid&B_{5}\\ \end{array} \right).$

It is not difficult to check that both $S_{4}$ and $S'_{4}$ generate $[85, 4, 64]$ codes, both $A_{4}$ and $B_{4}$ generate $[64, 4, 48]$ codes, both $S_{5}$ and $S'_{5}$ generate $[341, 5,256]$ codes, both $A_{5}$ and $B_{5}$ generate $[256, 5,192]$ codes, and $(\overline{B_{4}}\ B_{5})$ generates $[320, 5,240]$ code. The $[85, 4, 64]$ code, $[341, 5,256]$ code are called as quaternary simplex codes, the $[64, 4, 48]$ code, $[256, 5,192]$ code and $[320, 5,240]$ code are called as quaternary McDonald codes. All these codes are HSO codes.

If $A = A_{k, m}$ is a $k\times m$ matrix and the vectors formed by row linear combination of $A$ have largest weight $\delta$, then $A$ is called as an $(m, \delta)$ block. If $AA^{\dagger} = 0$, $A$ is called as an $(m, \delta)$ HSO block, where $A^{\dagger} = (A^{2})^{T}$ is the conjugate transpose of $A$.

Let $G = (\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n})$ be a $k\times n$ matrix, and $A_{j} = A_{k, m_{j}} = (\alpha_{j_{1}}, \cdots, \alpha_{j_{m_{j}}})$ be an $(m_{j}, \delta_{j})$ block of $G$ for $1\leq j\leq l$. Suppose $A_{i}$ and $A_{j}$ have index sets $I_{A_{i}} = \{ i_{1}, \cdots, i_{m_{i}}\}$ and $I_{A_{j}} = \{ j_{1}, \cdots, j_{m_{j}}\}$, if $I_{A_{i}}, I_{A_{j}}$ are disjoint. We say that $A_{i}$ and $A_{j}$ are disjoint. If any two $A_{i}$ and $A_{j}$ are disjoint for $1\leq j\leq l$, $A_{1}$, $A_{2}$, $\cdots$, $A_{l}$ are called disjoint blocks. It naturally follows that $l$ denotes the number of disjoint blocks in $G$.

Suppose $\mathcal{C}$ $= [n, k, d]$ is an HSO code with generator matrix $G$ and $G$ has a $k\times m$ sub-matrix $A$. If $A$ is an $(m, \delta)$ HSO block, then there is an $[n-m, k, d-\delta]$ HSO code, see ^[22].

Lemma 1. (^[22] Lemma 4) Let $N = N_{k} = \frac{4^{k}-1}{3}$. If there is an $[n, k, d]$ HSO code with $n\geq N+4$, then there are $[n-j, k, d-2\lceil\frac{j}{2}\rceil]$ HSO codes for $1\leq j\leq 4$.

Lemma 2. (^[22] Lemma 5) If $n = 256+m$, $m\geq 170$, and there is an $[m, 5, d_{5, m}]$ HSO code, then there are HSO codes with the following parameters: $[n, 5, d] = [256+m, 5,192+d_{5, m}]$, $[n-5i, 5,192+d_{5, m}-4i]$ for $i = 1, 2, 3, 4$, and $[n-5i-j, 5,192+d_{5, m}-4i-2\lceil\frac{j}{2}\rceil]$ for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

Let $N = N_{k} = \frac{4^{k}-1}{3}$. If there is an $[n, k, d]$ HSO code with $n\geq N+4$, then there are $[n-j, k, d-2\lceil\frac{j}{2}\rceil]$ HSO codes for $1\leq j\leq 4$.

Corollary 1. There are $[512-5i, 5,384-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, there are $[512-5i-j, 5,384-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

Denote the matrix formed by the last 64 columns of $S_{4}$ as $A_{4}$, and the matrix formed by the last 256 columns of $S_{5}$ as $A_{5}$.

3.   Proof of Theorem 1

In this section, Theorem 1 will be proved by constructing optimal HSO codes for $n \geq 492$, whose generator matrices can be obtained through removing some special HSO blocks from $G_{5, m}$ according to Lemmas 1 and 2, where $G_{5, m}$ means the generator matrix of an $[m, 5]$ code.

For $492\leq n \leq 1023$, according to the code lengths classification, we will discuss the following six cases, respectively.

Case a. $492\leq n\leq 597$.

Both $A_{5}$ and $B_{5}$ generate $[256, 5,192]$ HSO codes, and one can deduce that there are $[512-5i, 5,384-4i]$ HSO codes for $0\leq i\leq 4$, and $[512-5i-j, 5,384-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $0\leq i\leq 4$ and $1\leq j\leq 4$ according to Lemma 5 in ^[22].

Let $G_{5,576} = (A_{5}, \overline{B_{4}}, B_{5})$, and let $J$ be the $2\times 5$ all 2 matrix. We will show that $G_{5,576}$ has two $(21, 16)$ HSO blocks and eight $(5, 4)$ HSO blocks as follows.

Let $x^{T} = (2, 2)$. Define

$G_{5, 21} = \left(\begin{array}{ccccccc} 0_{5} & 1 & 1_{5} & 1_{5} & 1_{5}\\ S_{2} & 0&S_{2} & 2S_{2} & 3S_{2}\\ S_{2} & 0&S_{2} & 2S_{2} & 3S_{2}\\ \end{array} \right),$ $G'_{5, 21} = \left(\begin{array}{ccccccc} 0_{5} & 1 & 1_{5} & 1_{5} & 1_{5}\\ S_{2} & 0&S_{2} & 2S_{2} & 3S_{2}\\ 2S_{2}&x&J+2S_{2}&J+3S_{2}&J+S_{2}\\ \end{array} \right)$. $G_{5, 21} = \left(\begin{array}{ccccccc} 00000 & 1 11111 11111 11111\\ 01111 & 0 01111 02222 03333\\ 10123 & 0 10123 20231 30312\\ 01111 & 0 01111 02222 03333 \\ 10123 & 0 10123 20231 30312 \\ \end{array} \right),$ $G'_{5, 21} = \left(\begin{array}{ccccccc} 00000 & 1 11111 11111 1 1111\\ 01111 & 0 01111 02222 0 3333\\ 10123 & 0 10123 20231 3 0312\\ 02222 & 2 20000 21111 2 3333\\ 20231 & 2 02130 12130 3 2301\\ \end{array} \right).$

The first column of $G_{5, 21}$ is chosen from $A_{5}$, the second to the fifth columns of $G_{5, 21}$ are chosen from $\overline{B_{4}}$, and the last 16 columns are chosen from $B_{5}$. Similarly, the first column of $G'_{5, 21}$ is chosen from $A_{5}$, the second to the fifth columns of $G'_{5, 21}$ are chosen from $\overline{B_{4}}$, the sixth to the sixteenth columns and the twentieth column are chosen from $B_{5}$, the other four columns are chosen from $A_{5}$.

Using one of the columns $(00011)^{T}$, $(00012)^{T}$, $(00013)^{T}$, $(01001)^{T}$ from $A_{5}$ and a $4\times 5$ sub-matrix from $B_{5}$, one can construct four $(5, 4)$ HSO blocks as

Using a $4\times 5$ sub-matrix from $A_{5}$ and one of the columns $(01100)^{T}$, $(01200)^{T}$, $(01300)^{T}$, $(01110)^{T}$ from $\overline{B_{4}}$, one can construct four $(5, 4)$ HSO blocks as

It is not difficult to check that $G_{5,576}$ has two $(21, 16)$ HSO blocks, and eight $(5, 4)$ HSO blocks, and that these 10 blocks are disjoint. Let $G_{5,597} = (A_{5}, \overline{\overline{B_{3}}}, \overline{B_{4}}, B_{5})$. Then, $G_{5,576}$ is a sub-matrix of $G_{5,597}$ and $G_{5,597}$ generates a $[597, 5,448]$ HSO code.

Corollary 2. There are HSO codes with parameters as follows:

(1) $[597-5i, 5,448-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[597-5i-j, 5,448-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(2) $[576-5i, 5,432-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[576-5i-j, 5,432-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(3) $[555-5i, 5,416-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[555-5i-j, 5,416-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(4) $[534-5i, 5,400-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[534-5i-j, 5,400-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(5) $[512-5i, 5,384-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[512-5i-j, 5,384-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

Case b. $597\leq n\leq 682 = 341+341$.

Let $G_{5,682} = (S_{5}\ S'_{5}) = (\underline{\underline{S_{3}}}, \underline{A_{4}}, A_{5}\mid \overline{\overline{S_{3}}}, \overline{B_{4}}, B_{5})$. Then, $G_{5,576}$ is a sub-matrix of $G_{5,682}$. It is not difficult to check that $G_{5,682}$ has four $(21, 16)$ HSO block and eight $(5, 4)$ HSO blocks, and these 12 blocks are disjoint.

Corollary 3. There are HSO codes with parameters as follows:

(1) $[682-5i, 5,512-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[682-5i-j, 5,512-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(2) $[661-5i, 5,496-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[661-5i-j, 5,496-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(3) $[640-5i, 5,480-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[640-5i-j, 5,480-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(4) $[619-5i, 5,464-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[619-5i-j, 5,464-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

Case c. $683\leq n\leq 705$.

Let $G_{5,705} = (S_{5}\mid G_{5,364}) = (\underline{ S_{4}}, A_{5}\mid G_{5,364})$, then $A_{5}$ is a sub-matrix of $G_{5,705}$. It is not difficult to check that $A_{5}$ has four disjoint $(5, 4)$ HSO blocks.

Corollary 4. There are HSO codes with parameters as follows:

There are $[705-5i, 5,528-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[705-5i-j, 5,528-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$, and $[685-j, 5,512]$ HSO codes for $j = 1, 2$.

Case d. $706\leq n\leq 768$.

Let $\alpha = (0111)$, $\beta = (1023)$, $G_{5,768} = (A_{5}, D_{5}, B_{5})$, where

$D_{5} = \left(\begin{array}{cccccccccccccccccc} \alpha & \mathbf{0}_{21} & \mathbf{0}_{21} & \mathbf{0}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21} & \mathbf{1}_{21}\\ \mathbf{0}_{3\times 4}&S_{3}&S_{3}&S_{3}&S_{3} & 2S_{3} & 3S_{3}&S_{3} & 2S_{3} & 3S_{3}&S_{3} & 2S_{3} & 3S_{3}\\ \beta & \mathbf{1}_{21} & \mathbf{2}_{21} & \mathbf{3}_{21} & \mathbf{0}_{21} & \mathbf{0}_{21} & \mathbf{0}_{21} & \mathbf{2}_{21} & \mathbf{2}_{21} & \mathbf{2}_{21} & \mathbf{3}_{21} & \mathbf{3}_{21} & \mathbf{3}_{21}\\ \end{array} \right)$.

One can check that $D_{5}$ generates a $[256, 5,192]$ HSO code and $G_{5,768}$ generates a $[3\times 256, 5, 3\times 192]$ HSO code. From the construction of $D_{5}$, one can see that $D_{5}$ has a sub-matrix $\overline{B_{4}}$, hence $G_{5,576}$ is a sub-matrix of $G_{5,768}$. Thus, $G_{5,768}$ has two $(21, 16)$ HSO blocks and eight $(5, 4)$ HSO blocks, and these 10 blocks are disjoint.

Corollary 5. There are HSO codes with parameters as follows:

(1) $[768-5i, 5,576-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[768-5i-j, 5,576-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(2) $[747-5i, 5,560-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[747-5i-j, 5,560-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(3) $[726-5i, 5,544-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[726-5i-j, 5,544-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

Case e. $769\leq n\leq 832 = 512+320$.

Let $G_{5,832} = (A_{5}, \overline{B_{4}}, B_{5}, B_{5})$. Then, $G_{5,832}$ generates a $[832, 5,624]$ HSO code. It easy to see $G_{5,576}$ is a sub-matrix of $G_{5,832}$, hence $G_{5,832}$ has two $(21, 16)$ HSO blocks and eight $(5, 4)$ HSO blocks, and these 10 blocks are disjoint.

Corollary 6. There are HSO codes with parameters as follows:

(1) $[832-5i, 5,624-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[832-5i-j, 5,624-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(2) $[811-5i, 5,608-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[811-5i-j, 5,608-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

(3) $[790-5i, 5,592-4i]$ HSO codes for $i = 0, 1, 2, 3, 4$, and $[790-5i-j, 5,592-4i-2\lceil\frac{j}{2}\rceil]$ HSO codes for $i = 0, 1, 2, 3$ and $j = 1, 2, 3, 4$.

Case f. $833\leq n\leq 1023$.

Suppose $n = 341+n'$. Then, $492\leq n'\leq 682$. Let $G_{5, n} = (S_{5}\ G_{5, n'})$, where $G_{5, n'}$ is a generator matrix of an $[n', 5]$ optimal HSO code given in Cases a-e. Then, $G_{5, n} = (S_{5}\ G_{5, n'})$ generates an $[n = 341+n', 5]$ optimal HSO code.

Summarizing the above cases, $[n, 5]_{4}$ optimal HSO codes have been constructed for each $492\leq n \leq 1023$.

For $n\geq 1024$, denote $s = \lfloor\frac{n}{341}\rfloor$. Then, $s\geq3$. If $s\geq 3$, let $n = (s-2) \cdot 341+n''$ and $G_{5, n} = ((s-2)\cdot S_{5}, \ G_{5, n''})$. Then $G_{5, n}$ generates an $[n = (s-2)\cdot 341+n'', 5]$ optimal HSO code.

Summarizing the above discussions, we construct $[n, 5]$ HSO codes for each $n$ with $n\geq 492$. It follows that Theorem 1 holds.

4.   Conclusions

In this paper, we have verified $[n, 5, d_{so}(n, 5)]_{4}$ $= [n, 5, 2\lfloor \frac{d_{o}(n, 5)}{2}\rfloor]_{4}$ by constructing optimal HSO codes for each $n \geq 492$. When $492 \leq n \leq 1023$, the generator matrices of optimal HSO codes can be obtained by removing disjoint HSO blocks from some $G_{5, n}$. For $n \geq 1024$, optimal HSO codes can be constructed from corresponding ones with short length and some $[341, 5,256]_{4}$ quaternary simplex codes.

Author contributions

H. Song: Writing-original draft, writing-review and editing; Y. Ren: Conceptualization, methodology, software; R. Li: Conceptualization, writing-original draft, funding acquisition; Y. Liu: Data curation, writing-review and editing, funding acquisition. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (under Grant No. U21A20428), the Natural Science Foundation of Shaanxi Province (under Grant No. 2024JC-YBMS-055).

Conflict of interest

The authors declare no conflict of interest.