SYMPLECTIC QSD CODES AND THEIR
CLASSIFICATION OVER A NON-UNITAL
NON-LOCAL RING OF ORDER FOUR

Sarra Manseri, Adel Alahmadi, Widyan Bassafar, Rowena Alma Betty, Lucky Galvez, Patrick Solé

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Distributed under a Creative Commons Attribution 4.0 International License
SYMPLECTIC QSD CODES AND THEIR CLASSIFICATION
OVER A NON-UNITAL NON-LOCAL RING OF ORDER FOUR

SARRA MANSERI, ADEL ALAHMADI, WIDYAN BASSAFAR, ROWENA ALMA
BETTY, LUCKY GALVEZ and PATRICK SOLÉ

Abstract

In this article, we study the duality of codes over the non-unitary non-local ring

$$
H = \{a, b \mid 2a = 2b = 0, a^2 = 0, b^2 = b, ab = ba = 0\}
$$

under the symplectic inner product. That is, we introduce and characterized symplectic self-dual,
symplectic self-orthogonal and other notions of duality of codes in this ring. A classification of symplectic
quasi self-dual codes over $\mathcal{H}$ is also given for short lengths.

2010 Mathematics subject classification: 94B05, 16D10.

Keywords and phrases: Symplectic inner product, self-orthogonal codes, QSD codes, self-dual codes, nice codes.

1. Introduction

Self dual codes have been widely studied over finite fields [14, 18, 30] and finite rings
[15, 17, 22]. Recently, this trend in research has been extended to quasi self-dual
codes (QSD) over non-unitary rings, primarily concentrating on commutative non-
unitary rings of order four [7, 8, 10]. Furthermore, Kushwaha et al. [19] established
QSD codes over a non-unitary non-commutative ring with nine elements. Moreover,
three propagation rules to generate self-orthogonal, QSD and self-dual codes over non-
unitary rings of order 9 are given in [1, 2]. In [24, 26], self-orthogonal, QSD and self-
dual codes over non-unitary rings with respect to the symplectic inner product were
introduced and studied. Linear complementary dual (LCD) codes under the Euclidean
and symplectic inner products over non-unitary rings have also been considered in
[6, 19, 25, 29].

In recent years, the classification of linear codes over finite fields and non-unitary
rings has received much attention [3, 20, 23]. In [7, 10], Alahmadi et al. gave a
complete classification of self-orthogonal, QSD and self-dual codes for short lengths
and small types over non-unitary rings of order 4. The classification for small lengths
and types of codes over non-unitary rings of odd characteristic was given in [4, 5].
The current literature on QSD, self-dual and LCD codes over non-unitary rings and their classification, and the available research gap on such codes, especially under the symplectic inner product, has motivated this study of codes over the non-unitary non-local ring $H$ of order four defined by generators and relations as

$$H = \{a, b \mid 2a = 2b = 0, a^2 = 0, b^2 = b, ab = ba = 0\}.$$

In particular, we introduce and provide construction of self-orthogonal, QSD and self-dual codes over this ring with respect to the symplectic inner product. Then, we derive symplectic LCD and symplectic nice codes. We show that over the ring $H$, the minimum symplectic distance of a linear code does not exceed the minimum symplectic distance of any of its associated binary codes. The relationship between minimum Hamming and symplectic distances of a linear code over $H$ is also determined. Finally, we classify symplectic QSD codes over $H$ of length at most 8.

This paper is organized as follows. The next section contains preliminaries and basic definitions needed in the subsequent sections. Section 3 characterizes symplectic self-orthogonal, symplectic QSD and symplectic self-dual codes over $H$. We define the minimum symplectic distance of linear codes over $H$ of length $n = 2m$ and give its relation to the minimum Hamming distance. We also study symplectic nice and symplectic LCD codes over $H$. Section 4 classifies symplectic QSD codes over $H$ for small lengths. Section 5 concludes this article.

## 2. Preliminaries

In this section, basic concepts about codes will be reviewed and important definitions regarding the ring $H$ and codes over this ring will be introduced. From here onwards, let $n$ be an even integer such that $n = 2m$ for some integer $m$.

**2.1. Binary codes** For non-negative integers $n$ and $k$, an $[n, k]$ linear code $C$ over $\mathbb{F}_2$, also known as a binary $[n, k]$ code, is a $k$-dimensional subspace of $\mathbb{F}_2^n$. Denote by $\mathbf{x} = (\mathbf{u}|\mathbf{v})$ and $\mathbf{y} = (\mathbf{w}|\mathbf{t})$ the juxtaposition of the vectors $\mathbf{u}, \mathbf{v} \in \mathbb{F}_2^m$, and $\mathbf{w}, \mathbf{t} \in \mathbb{F}_2^m$, respectively. The **symplectic inner product** of $\mathbf{x}$ and $\mathbf{y}$ is defined as

$$\langle \mathbf{x}, \mathbf{y} \rangle_S = \langle \mathbf{u}, \mathbf{t} \rangle_E + \langle \mathbf{v}, \mathbf{w} \rangle_E = \mathbf{x} \Omega \mathbf{y}^T,$$

where $\langle \cdot, \cdot \rangle_E$ is the (usual) Euclidean inner product, $\Omega = \begin{bmatrix} 0_m & I_m \\ I_m & 0_m \end{bmatrix}$, $0_m$ is the $m \times m$ zero matrix, and $I_m$ is the identity matrix of order $m$. The **symplectic dual** of a binary $[n, k]$ code $C$ is defined as

$$C^{\perp s} = \{ \mathbf{x} \in \mathbb{F}_2^n : \langle \mathbf{x}, \mathbf{y} \rangle_S = 0, \forall \mathbf{y} \in C \}.$$

It is easy to see that $C^{\perp s}$ is a binary $[n, n-k]$-code.

A binary code $C$ is said to be **symplectic self-orthogonal** (respectively, **symplectic self-dual**) if $C \subseteq C^{\perp s}$ (respectively, $C = C^{\perp s}$). If $C \cap C^{\perp s} = \{0\}$, we say that $C$ is a **symplectic LCD** (or **symplectic linear complementary dual**) code.
For any binary vector $\mathbf{y} = (\mathbf{u}|\mathbf{v})$ of length $n$, where $\mathbf{u} = (u_1, u_2, \dots, u_m)$ and $\mathbf{v} = (v_1, v_2, \dots, v_m)$, the **symplectic weight** $\omega_S(\mathbf{y})$ is defined as

$$ \omega_S(\mathbf{y}) = |\{i : (u_i, v_i) \neq (0, 0), 1 \le i \le m\}|. $$

The **minimum symplectic distance** of a binary linear code $C$ is

$$ d_S(C) = \min\{\omega_S(\mathbf{x}) \mid \mathbf{0} \neq \mathbf{x} \in C\}. $$

If $d_H(C)$ denotes the minimum Hamming distance of $C$, then we have the inequalities from [21] given by

$$ \left\lfloor \frac{d_H(C)}{2} \right\rfloor \le d_S(C) \le \min \left\{ d_H(C), \left\lfloor \frac{n-k+2}{2} \right\rfloor \right\}. \quad (2.1) $$

**2.2. Codes over the ring H** As in [12], the ring $H$ on two generators $a$ and $b$ is defined according to the presentation

$$ H = \langle a, b \mid 2a = 2b = 0, a^2 = 0, b^2 = b, ab = ba = 0 \rangle. $$

The ring $H$ has four elements $\{0, a, b, c\}$ where $c = a + b$ and the addition and multiplication tables are as follows.

$$ \begin{array}{c|llll} + & 0 & a & b & c \\ \hline 0 & 0 & a & b & c \\ a & a & 0 & c & b \\ b & b & c & 0 & a \\ c & c & b & a & 0 \end{array} $$

$$ \begin{array}{c|llll} \times & 0 & a & b & c \\ \hline 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & 0 \\ b & 0 & 0 & b & b \\ c & 0 & 0 & b & b \end{array} $$

Observe that $H$ is a commutative ring of characteristic 2 and has no identity element for multiplication. Moreover, it is a semi-local ring with two maximal ideals $m_a = \{0, a\}$ and $m_b = \{0, b\}$, and $H = m_a \oplus m_b$. We define a natural action of $\mathbb{F}_2$ on $H$ by the rule $h0 = 0h = 0$ and $h1 = 1h = h$, for all $h \in H$. This action is distributive in the sense that for $h \in H$, $h(s + t) = hs + ht$, where $s, t \in \mathbb{F}_2$ and the sum $s + t$ is carried out modulo 2.

A **linear code** over $H$, or simply an $H$-code, of length $n$ is an $H$-submodule of the $H^n$ module. Define the map $\psi_H$ from $H$ to $\mathbb{F}_4 := \mathbb{F}_2[\omega]$, where $\omega^2 + \omega = 1$, by the substitution

$$ 0 \mapsto 0, \quad a \mapsto \omega, \quad b \mapsto 1, \quad c \mapsto \omega^2. $$

This maps extents naturally to $H^n$. Hence, for every $H$-code $C$ of length $n$, there is an additive $\mathbb{F}_4$ code $\psi_H(C)$.

Moreover, let $\alpha_a : H \to \mathbb{F}_2$ be the map of reduction modulo $m_a$, and $\alpha_b : H \to \mathbb{F}_2$ be the map of reduction modulo $m_b$. These maps are defined respectively as

$$ \alpha_b(0) = \alpha_b(b) = 0, \quad \alpha_b(a) = \alpha_b(c) = 1, \text{ and } \alpha_a(0) = \alpha_a(a) = 0, \quad \alpha_a(b) = \alpha_a(c) = 1. $$
This map again extends to $H^n$. Therefore, with an $H$-code $C$, we associate two binary codes $C_a = \alpha_b(C)$ and $C_b = \alpha_a(C)$. With this, $C$ can be written as the direct sum $C = aC_a \oplus bC_b$. See [6, 9] for more details on the ring $H$ and linear codes over $H$.

The **symplectic inner product** of $\mathbf{x} = (\mathbf{u}|\mathbf{v})$ and $\mathbf{y} = (\mathbf{w}|\mathbf{t})$, where $\mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{t} \in H^m$ is given by

$$ \langle \mathbf{x}, \mathbf{y} \rangle_S = \langle \mathbf{u}, \mathbf{t} \rangle_E + \langle \mathbf{v}, \mathbf{w} \rangle_E, $$

where $\langle \cdot, \cdot \rangle_E$ denotes the (usual) Euclidean inner product over $H$. The **symplectic dual** of an $H$-code $C$ is given by

$$ C^{\perp s} = \{ \mathbf{y} \in H^n : \langle \mathbf{y}, \mathbf{x} \rangle_S = 0, \forall \mathbf{x} \in C \}. $$

As in [6], an $H$-code $C$ is said to be **symplectic self-orthogonal** if $C \subseteq C^{\perp s}$ and **symplectic self-dual** if $C = C^{\perp s}$. Furthermore, we say that an $H$-code $C$ of length $n$ is **symplectic nice** if it satisfies $|C||C^{\perp s}| = 4^n$. If $C$ is symplectic self-orthogonal with cardinality $4^m$, we also say that $C$ is **symplectic quasi self-dual** or **symplectic QSD**. Finally, $C$ is **symplectic LCD** if $C \cap C^{\perp s} = \{0\}$.

Every two $H$-codes are said to be **permutation equivalent** if there exists a permutation of coordinates that maps one to the other. If these codes are either both symplectic self-orthogonal or both symplectic LCD, we say that these codes are **symplectic equivalent**.

### 3. Symplectic Duality

First, we characterize the symplectic dual of an $H$-code $C$ of length $n$.

**THEOREM 3.1.** Let $C$ be an $H$-code of length $n$ such that $C = aC_a \oplus bC_b$. Then the symplectic dual of $C$ can be expressed uniquely as

$$ C^{\perp s} = a\mathbb{F}_2^n \oplus bC_b^{\perp s}. $$

**PROOF.** Let $\mathbf{c} \in C^{\perp s}$ and write $\mathbf{c} = a\mathbf{x} + b\mathbf{y}$ where $\mathbf{x}, \mathbf{y} \in \mathbb{F}_2^n$. Suppose $\mathbf{z} \in C_b$. Since $bC_b \subseteq C$, we have $b\mathbf{z} \in C$ and

$$ 0 = \langle \mathbf{c}, b\mathbf{z} \rangle_S = \langle a\mathbf{x} + b\mathbf{y}, b\mathbf{z} \rangle_S = b\langle \mathbf{y}, \mathbf{z} \rangle_S, $$

which means $\langle \mathbf{y}, \mathbf{z} \rangle_S = 0$. Thus, $\mathbf{y} \in C_b^{\perp s}$ and we have $C^{\perp s} \subseteq a\mathbb{F}_2^n \oplus bC_b^{\perp s}$.

Conversely, suppose $\mathbf{c} \in a\mathbb{F}_2^n \oplus bC_b^{\perp s}$ and write $\mathbf{c} = a\mathbf{x} + b\mathbf{y}$ where $\mathbf{x} \in \mathbb{F}_2^n, \mathbf{y} \in C_b^{\perp s}$. Let $\mathbf{c}' = a\mathbf{t} + b\mathbf{w} \in C$. Then $\mathbf{t} \in C_a, \mathbf{w} \in C_b$. Observe that

$$ \langle \mathbf{c}, \mathbf{c}' \rangle_S = \langle a\mathbf{x} + b\mathbf{y}, a\mathbf{t} + b\mathbf{w} \rangle_S = b\langle \mathbf{y}, \mathbf{w} \rangle_S = b \cdot 0 = 0. $$

This implies that $\mathbf{c} \in C^{\perp s}$ and so $a\mathbb{F}_2^n \oplus bC_b^{\perp s} \subseteq C^{\perp s}$.

We now have $C^{\perp s} = a\mathbb{F}_2^n + bC_b^{\perp s}$ and since $a\mathbb{F}_2^n \cap bC_b^{\perp s} = \{0\}$, the sum is direct. $\square$

**COROLLARY 3.2.** Suppose $C = aC_a \oplus bC_b$ is an $H$-code of length $n$. Then $(C^{\perp s})^{\perp s} = C$ if and only if $C_a = \mathbb{F}_2^n$.
**PROOF.** From Theorem 3.1, $(C^{\perp s})^{\perp s} = a\mathbb{F}_2^n + bC_b$. Since $C = aC_a \oplus bC_b$, we conclude that $(C^{\perp s})^{\perp s} = C$ if and only if $C_a = \mathbb{F}_2^n$. □

In the following results, we characterized the symplectic nice, symplectic self-orthogonal and symplectic QSD codes over $H$.

**THEOREM 3.3.** An $H$-code $C = aC_a \oplus bC_b$ of length $n$ is symplectic nice if and only if $C_a = \{0\}$.

**PROOF.** Let $k_a$ and $k_b$ be the dimensions of $C_a$ and $C_b$, respectively. By Theorem 3.1, we have

$$|C||C^{\perp s}| = |C_a||C_b||\mathbb{F}_2^n||C_b^{\perp s}| = 2^{k_a}2^{k_b}2^n2^{n-k_b} = 2^{2n+k_a}.$$

Hence, $|C||C^{\perp s}| = 4^n$ if and only if $k_a = 0$. Equivalently, $C$ is symplectic nice if and only if $C_a = \{0\}$. □

**THEOREM 3.4.** An $H$-code $C = aC_a \oplus bC_b$ is symplectic self-orthogonal if and only if $C_b$ is a binary symplectic self-orthogonal code.

**PROOF.** Let $\mathbf{c}, \mathbf{c}' \in C$ and write $\mathbf{c} = ax + by$ and $\mathbf{c}' = ax' + by'$, where $x, x' \in C_a$ and $y, y' \in C_b$. Since $C$ is symplectic self-orthogonal,

$$0 = \langle \mathbf{c}, \mathbf{c}' \rangle_S = \langle ax + by, ax' + by' \rangle_S = b\langle y, y' \rangle_S.$$

This means $\langle y, y' \rangle_S = 0$ and the result follows. □

The following results show that every symplectic QSD and symplectic self-dual $H$-code is determined by its associated binary codes.

**THEOREM 3.5.** Let $C = aC_a \oplus bC_b$ be an $H$-code of length $n$. Then $C$ is a symplectic QSD code if and only if both of the following two conditions hold:

1. $C_b$ is a binary symplectic self-orthogonal $[n, k]$ code, and

2. $C_a$ is a binary $[n, n-k]$ code.

**PROOF.** By Theorem 3.4, the code $C$ is symplectic self-orthogonal if and only if $C_b$ is binary symplectic self-orthogonal code. Moreover, since $|C| = |C_a||C_b|$, we have $|C| = 2^n$ if and only if the binary codes $C_a$ and $C_b$ have complementary dimensions. □

**THEOREM 3.6.** Let $C = aC_a \oplus bC_b$ be an $H$-code of length $n$. Then $C$ is a symplectic self-dual if and only if both of the following conditions hold:

1. $C_a = \mathbb{F}_2^n$, and

2. $C_b$ is a binary symplectic self-dual code of length $n$.

**PROOF.** By Theorem 3.1 and the definition of symplectic self-dual codes, the result is obtained. □

The following result shows that any $H$-code cannot be both symplectic self-dual and symplectic QSD.
**PROPOSITION 3.7.** Let $\mathcal{D}$ be the set of all symplectic self-dual H-codes of length $n$ and $\mathcal{S}$ be the set of all symplectic QSD H-codes of length $n$. Then $\mathcal{D} \cap \mathcal{S} = \emptyset$.

**PROOF.** Let $C$ be an $H$-code of length $n$. Assume $C$ is both symplectic QSD and symplectic self-dual. Then, $|C| = |C^{\perp s}| = 4^m$. Furthermore, we have $|C||C^{\perp s}| = 4^n$, which implies that $C$ is symplectic nice. By Theorem 3.3, $C_a = \{0\}$. Thus, this leads to a contradiction to Theorem 3.6. Hence, there is no such code $C$. $\square$

A fundamental concept in coding theory is the minimum distance of a code. For an arbitrary $H$-code $C$, we denote by $d_S(C)$ the minimum symplectic distance of $C$. Next, it will be shown that the minimum symplectic distance can be determined from the associated binary codes.

**THEOREM 3.8.** Let $C = aC_a \oplus bC_b$ be an $H$-code, where $C_a$ and $C_b$ are nonzero binary codes. Then

$$d_S(C) = \min\{d_S(C_a), d_S(C_b)\},$$

where $d_S(C_a)$ and $d_S(C_b)$ are the minimum symplectic distances of the binary codes $C_a$ and $C_b$, respectively.

**PROOF.** From the definitions of $d_S(C_a)$ and $d_S(C_b)$, there exist nonzero binary vectors $\mathbf{x} \in C_a, \mathbf{y} \in C_b$ such that $d_S(C_a) = \omega_S(\mathbf{x})$ and $d_S(C_b) = \omega_S(\mathbf{y})$. Since $aC_a \subseteq C$ and $bC_b \subseteq C$, we obtain $a\mathbf{x}, b\mathbf{y} \in C$. Observe that $\omega_S(a\mathbf{x}) = \omega_S(\mathbf{x})$ and $\omega_S(b\mathbf{y}) = \omega_S(\mathbf{y})$. These imply that $d_S(C) \le \min\{d_S(C_a), d_S(C_b)\}$.

It remains to show that $d_S(C) \ge \min\{d_S(C_a), d_S(C_b)\}$. Let $\mathbf{z} \in C$ such that $d_S(C) = \omega_S(\mathbf{z})$ and write $\mathbf{z} = a\mathbf{z}_1 + b\mathbf{z}_2$ where $\mathbf{z}_1 \in C_a, \mathbf{z}_2 \in C_b$. Then we have the following cases:

* If $\mathbf{z}_1 = \mathbf{0}, \mathbf{z}_2 \ne \mathbf{0}$, then $\omega_S(\mathbf{z}) = \omega_S(b\mathbf{z}_2) = \omega_S(\mathbf{z}_2) \ge d_S(C_b)$.

* If $\mathbf{z}_1 \ne \mathbf{0}, \mathbf{z}_2 = \mathbf{0}$, then $\omega_S(\mathbf{z}) = \omega_S(a\mathbf{z}_1) = \omega_S(\mathbf{z}_1) \ge d_S(C_a)$.

* If $\mathbf{z}_1 \ne \mathbf{0}, \mathbf{z}_2 \ne \mathbf{0}$, then $\omega_S(\mathbf{z}) \ge \omega_S(b\mathbf{z}) = \omega_S(b\mathbf{z}_2) = \omega_S(\mathbf{z}_2) \ge d_S(C_b)$.

In each case, $d_S(C) = \omega_S(\mathbf{z}) \ge \min\{d_S(C_a), d_S(C_b)\}$. As a result, we have $d_S(C) = \min\{d_S(C_a), d_S(C_b)\}$. $\square$

From Theorem 3.1 and Theorem 3.6, we observe that all symplectic self-dual $H$-codes have the same minimum symplectic distance equal to 1. Furthermore, by applying the inequality in (2.1), Theorem 3.6, and Theorems 20 and 23 in [6], we deduce the next remark.

**REMARK 3.9.** Let $C$ be an $H$-code of length $n$. If $C$ is a (Euclidean) self-dual code over $H$, then

$$d_S(C) = d_H(C) = 1,$$

where $d_H(C)$ is the minimum (Hamming) distance of $C$ and $d_S(C)$ is the minimum symplectic distance of $C$.
An additive subgroup of $\mathbb{F}_4^n$ with $2^k$ elements is called an $(n, 2^k)$ **additive code** over $\mathbb{F}_4$ of length $n$. Its a free $\mathbb{F}_2$-module with $2^k$ codewords, for some $0 \le k \le 2n$. The trace map $Tr: \mathbb{F}_4 \to \mathbb{F}_2$ is defined as $Tr(x) = x + x^2$, and can be extended naturally from $\mathbb{F}_4^n$ to $\mathbb{F}_2^n$. The **trace inner product** $\langle \mathbf{x}, \mathbf{y} \rangle_T$ of vectors $\mathbf{x}, \mathbf{y} \in \mathbb{F}_4^n$ is defined as

$$ \langle \mathbf{x}, \mathbf{y} \rangle_T = \mathrm{Tr} \left( \langle \mathbf{x}, \mathbf{y}^2 \rangle_E \right) = \mathrm{Tr} \left( \sum_{i=1}^{n} x_i y_i^2 \right). $$

Let $C$ be an additive code of length $n$ over $\mathbb{F}_4$. The **trace dual** of $C$ is defined as

$$ C^{\perp T} = \{ \mathbf{y} \in \mathbb{F}_4^n : \langle \mathbf{x}, \mathbf{y} \rangle_T = 0, \forall \mathbf{x} \in C \}. $$

If $C$ is an $(n, 2^k)$ additive code, then its trace dual is an $(n, 2^{2n-k})$ additive code. We say $C$ is **trace self-orthogonal** if $C \subseteq C^{\perp T}$ and **trace self-dual** if $C^{\perp T} = C$. Moreover, $C$ is called **additive with complementary dual (ACD)** code if $C \cap C^{\perp T} = \{0\}$. Interestingly, the image of any symplectic self-dual $H$-code under the map $\psi_H$ can never be a trace self-dual code over $\mathbb{F}_4$. However, Proposition 8 in [6] asserts that the trace dual of the image of this code under $\psi_H$ is a trace self-orthogonal code over $\mathbb{F}_4$.

**PROPOSITION 3.10.** Let $C = aC_a \oplus bC_b$ be an $H$-code of length $n$. If $C$ is a symplectic self-dual code, then $\psi_H(C)$ is not trace self-dual. Moreover, $\psi_H(C)^{\perp T}$ have cardinality equal to $2^m$.

**PROOF.** From the symplectic self-duality of $C$, Theorems 3.1 and 3.6, we obtain $|C_a| = |\mathbb{F}_2^n| = 2^{2m}$ and $|C_b| = 2^m$. Moreover, $|\psi_H(C)| = |C| = 2^{3m}$, and $|\psi_H(C)^{\perp T}| = 2^{4m-3m} = 2^m$. Since $\psi_H(C)$ and $\psi_H(C)^{\perp T}$ do not have the same cardinality, they are not the equal. Hence, $\psi_H(C)$ is not a trace self-dual code over $\mathbb{F}_4$. $\square$

The next theorem follows from Theorem 21 in [6]. It establishes the symplectic equivalence of two $H$-codes in terms of their associated binary codes.

**THEOREM 3.11.** Suppose $C = aC_a \oplus bC_b$ and $D = aD_a \oplus bD_b$ are symplectic self-dual $H$-codes of length $n$. Then $C$ and $D$ are symplectic equivalent if and only if $C_b$ and $D_b$ are symplectic equivalent.

**PROOF.** Since $C$ and $D$ are symplectic self-dual, by Theorem 3.6, $C_a = D_a = \mathbb{F}_2^n$ and $C_b$ and $D_b$ are binary symplectic self-dual codes. Now, two codes $C$ and $D$ are symplectic equivalent if there exists a permutation matrix $\mathcal{P}$ such that $D = C\mathcal{P}$. Moreover, $D_b = \alpha_a(D) = \alpha_a(C\mathcal{P}) = \alpha_a(C)\mathcal{P} = C_b\mathcal{P}$. Thus, $C_b$ and $D_b$ are symplectic equivalent.

Conversely, suppose $C_b$ and $D_b$ are symplectic equivalent. Then there exists a permutation matrix $\mathcal{P}$ such that $D_b = C_b\mathcal{P}$. Observe that $\mathbb{F}_2^n = \mathbb{F}_2^n\mathcal{P}$. By applying again Theorem 3.6, we obtain

$$ D = a\mathbb{F}_2^n \oplus bD_b = a\mathbb{F}_2^n\mathcal{P} \oplus bC_b\mathcal{P} = C\mathcal{P}. $$

Hence, $D$ and $C$ are symplectic equivalent. $\square$
We conclude this section by the following results on symplectic LCD H-codes.

**THEOREM 3.12.** An H-code $C = aC_a \oplus bC_b$ of length $n$ is a symplectic LCD code if and only if $C$ is symplectic nice and $C_b$ is a binary symplectic LCD code.

**PROOF.** Suppose $C$ is a symplectic LCD code. From the definition and Theorem 3.1, $C_a \cap \mathbb{F}_2^n = \{0\}$ and $C_b \cap C_b^{\perp s} = \{0\}$. Therefore, $C_a$ is the zero code and $C_b$ is a binary symplectic LCD code. Moreover, by Theorem 3.3, the code $C$ is symplectic nice.

Conversely, suppose $C$ is symplectic nice and $C_b$ is a binary symplectic LCD code. By Theorem 3.3, $C_a = \{0\}$. Therefore, $C_a \cap \mathbb{F}_2^n = \{0\}$ and $C_b \cap C_b^{\perp s} = \{0\}$. From Theorem 3.1, $C \cap C_b^{\perp s} = \{0\}$, which proves that $C$ is a symplectic LCD code. $\square$

The image under the map $\psi_H$ of every symplectic LCD H-code is a trace self orthogonal code over $\mathbb{F}_4$. Note that it cannot be an ACD code over $\mathbb{F}_4$. These are similar results with Euclidean LCD codes (see Proposition 9 in [6]).

**PROPOSITION 3.13.** Let $C = aC_a \oplus bC_b$ be an nonzero symplectic LCD H-code of length $n$. Then $\psi_H(C)$ is an additive trace self-orthogonal code over $\mathbb{F}_4$. Specifically, $\psi_H(C)$ is not an ACD code over $\mathbb{F}_4$.

**PROOF.** Since $C$ is a symplectic LCD code, by Theorem 3.3 and Theorem 3.12, $C_a = \{0\}$. So $C = bC_b$. Thus, $\psi_H(C) = C_b$ and $\psi_H(C)^{\perp T} = C_b^{\perp T}$. Let $\mathbf{u}, \mathbf{v} \in C_b$. We note that

$$ \langle \mathbf{u}, \mathbf{v} \rangle_T = \mathrm{Tr}(\langle \mathbf{u}, \mathbf{v}^2 \rangle_E) = \mathrm{Tr}(\langle \mathbf{u}, \mathbf{v} \rangle_E) $$

since each component of the vectors $\mathbf{u}$ and $\mathbf{v}$ is either 0 or 1. By definition of the trace map, $\langle \mathbf{u}, \mathbf{v} \rangle_T = 0$, which implies that $C_b$ is trace self-orthogonal. Hence, $\psi_H(C) \subseteq \psi_H(C)^{\perp T}$. Since $C$ is a nonzero code, $\psi_H(C) \cap \psi_H(C)^{\perp T} \neq \{0\}$. Consequently, $\psi_H(C)$ is an additive self-orthogonal code but not an ACD code. $\square$

For the equivalence between two symplectic LCD codes over $H$, we have the following results.

**COROLLARY 3.14.** Let $C = aC_a \oplus bC_b$ and $D = aD_a \oplus bD_b$ be two symplectic LCD H-codes. Then $C$ and $D$ are symplectic equivalent if and only if $C_b$ and $D_b$ are symplectic equivalent.

**PROOF.** Since $C$ and $D$ are symplectic LCD codes, by Theorem 3.12 and Theorem 3.3, $C_a$ and $D_a$ are zero codes and $C_b$ and $D_b$ are binary symplectic LCD codes. From Theorem 21 in [6], $C$ and $D$ are symplectic equivalent if and only if $C_b$ and $D_b$ are symplectic equivalent. $\square$
## 4. Classification and Numerical Results

As an analogue to the classification method of QSD H-codes of length $n$ discussed in [9] and the classification of self-dual codes over $\mathbb{Z}_{rs}$ where $r$ and $s$ are relatively prime in [27], the classification can be accomplished by double coset decompositions of $S_n$ by automorphism groups of binary self-orthogonal symplectic code $C_b$ and a binary code $C_a$ of complementary dimension. Note that the permutation of coordinates does not generally preserve symplectic self-orthogonality. Therefore, the automorphism group of a binary self-orthogonal symplectic code $C_b$ consists of all permutations that fixes $C_b$ and preserves symplectic self-orthogonality. We have to find all symplectic QSD codes that are symplectic equivalent to $aC_a + bC_b$ for a given pair $(C_a, C_b)$. Similarly, we write SDR for System of Distinct Representatives, where its elements are representative of subsets (here the double cosets) in a set (here the group $S_n$) partition. We have the following results as an analogue of Theorem 2 in [9].

**THEOREM 4.1.** Let $(C_a, C_b)$ be a pair of codes as defined in Theorem 3.5. Then the set

$$S_{C_a, C_b} := \{aC_a + b\sigma(C_b) \mid \sigma \text{ runs over an SDR of } \mathrm{Aut}(C_a)\setminus S_n/\mathrm{Aut}(C_b)\}$$

forms a set of inequivalent symplectic QSD H-codes of length $n$. Moreover, $|S_{C_a, C_b}| = |\mathrm{Aut}(C_a)\setminus S_n/\mathrm{Aut}(C_b)|$.

As a consequence of Theorem 3.6 and Theorem 4.1, we have the following corollary, which is similar to Corollary 3 in [9].

**COROLLARY 4.2.** Let $L_b$ be the set of all inequivalent binary symplectic self-orthogonal $[n, k]$ codes and $L_a$ be the set of all inequivalent binary $[n, n-k]$ codes. Then the set of all symplectic QSD H-codes of length $n$ is the disjoint union

$$\bigcup_{\substack{C_a \in L_a \\ C_b \in L_b}} S_{C_a, C_b}.$$

To classify symplectic QSD H-codes, we follow the following classification algorithm. Given a length $n \ge 2$ and $k \le m$, we do the following:

1. Write a list $L_b$ of binary symplectic self-orthogonal $[n, k]$ codes;

2. Write a list $L_a$ of binary $[n, n-k]$ codes;

3. For every pair $(C_a, C_b) \in L_a \times L_b$,

(a) we compute the automorphism groups $\mathrm{Aut}(C_a)$ and $\mathrm{Aut}(C_b)$,

(b) find a list $\sigma_1, \dots, \sigma_r$ of representatives of $\mathrm{Aut}(C_a)\setminus S_n/\mathrm{Aut}(C_b)$, and

(c) for $i = 1$ up to $r$, compute $aC_a + b\sigma_i(C_b)$.

The number of codes in $L_a$ is given in [13]. To obtain the list $L_b$, the following lemmas will be helpful.

**LEMMA 4.3.** ([16, Lemma 3.1]) Every vector in $\mathbb{F}_2^n$ is orthogonal to itself with symplectic inner product.
By Lemma 4.3, there are $n$ inequivalent binary symplectic self-orthogonal $[n, 1]$-codes.

**LEMMA 4.4.** ([16, Lemma 3.2]) *The number of binary symplectic self-orthogonal codes of length $n$ and dimension $k$ is given by*

$$ \Phi_{\mathbb{F}_2}(n, k) = \frac{(2^{n-2k+2} - 1)(2^{n-2k+4} - 1) \cdots (2^n - 1)}{(2^k - 1)(2^{k-1} - 1) \cdots (2 - 1)}. $$

Observe that Lemma 4.4 does not hold for $k = 0$. Clearly, the zero code of length $n$ is a binary symplectic self-orthogonal code. Note that the additive generator matrix of each of the codes is obtained by taking the sums of the rows. All computer calculations were done using MAGMA [11].

**4.1. Length 2** There are six inequivalent symplectic QSD $H$-codes of length 2. For $k = 0$, we have the QSD code $a\mathbb{F}_2^2$. For $k = 1$, we obtained QSD codes with additive generator matrices given by

$$ \left\{ \begin{bmatrix} a & 0 \\ b & 0 \end{bmatrix}, \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}, \begin{bmatrix} a & 0 \\ b & b \end{bmatrix}, \begin{bmatrix} a & a \\ b & 0 \end{bmatrix} \begin{bmatrix} a & a \\ b & b \end{bmatrix} \right\}. $$

In this case, $L_a = L_b$, where the binary codes have generator matrices given by the set $\{[1 \ 0], [1 \ 1]\}$.

**4.2. Length 4** There are 92 inequivalent symplectic QSD $H$-codes of length 4. For $k = 0$, we have the QSD code $a\mathbb{F}_2^4$. For $k = 1$, there are four inequivalent binary $[4, 3]$ codes in $L_a$ and there are four inequivalent binary symplectic self-orthogonal $[4, 1]$ codes in $L_b$. These result to 26 inequivalent symplectic QSD $H$-codes. For $k = 2$, there are six inequivalent binary $[4, 2]$ codes in $L_a$ and there are four inequivalent binary symplectic self-orthogonal $[4, 2]$ codes in $L_b$. From these, we obtained 65 inequivalent symplectic QSD $H$-codes. We summarize the results in the following table.

<table><caption>**TABLE 1.** The number of symplectic QSD $H$-codes of length 4</caption><thead><tr><th>$k$</th><th>Number of codes in $L_a$</th><th>Number of codes in $L_b$</th><th>Number of $H$-codes</th></tr></thead><tbody><tr><td>1</td><td>4</td><td>4</td><td>26</td></tr><tr><td>2</td><td>6</td><td>4</td><td>65</td></tr></tbody></table>

**4.3. Length 6** There are 4,035 inequivalent symplectic QSD $H$-codes of length 6. For $k = 0$, we have the QSD code $a\mathbb{F}_2^6$. For $k \ge 1$, we summarize the results in the following table.
<table><caption>TABLE 2. The number of symplectic QSD H-codes of length 6</caption><thead><tr><td>k</td><td>Number of codes in L<sub>a</sub></td><td>Number of codes in L<sub>b</sub></td><td>Number of H-codes</td></tr></thead><tbody><tr><td>1</td><td>6</td><td>6</td><td>71</td></tr><tr><td>2</td><td>16</td><td>14</td><td>1,483</td></tr><tr><td>3</td><td>22</td><td>10</td><td>2,480</td></tr></tbody></table>
**4.4. Length 8** There are 870,069 inequivalent symplectic QSD *H*-codes of length 8. For *k* = 0, we have the QSD code *aF*<sub>2</sub><sup>8</sup>. For *k* ≥ 1, refer to the following table.

<table><caption>TABLE 3. The number of symplectic QSD *H*-codes of length 8</caption><thead><tr><td><i>k</i></td><td>Number of codes in <i>L</i><sub><i>a</i></sub></td><td>Number of codes in <i>L</i><sub><i>b</i></sub></td><td>Number of <i>H</i>-codes</td></tr></thead><tbody><tr><td>1</td><td>8</td><td>8</td><td>148</td></tr><tr><td>2</td><td>32</td><td>30</td><td>14,345</td></tr><tr><td>3</td><td>77</td><td>62</td><td>386,814</td></tr><tr><td>4</td><td>106</td><td>28</td><td>468,761</td></tr></tbody></table>

We found more inequivalent symplectic QSD codes over *H* for even lengths compared to the number of inequivalent QSD codes (under the Euclidean inner product) listed in [10].

**5. Conclusion**

In this article, symplectic (self-orthogonal, QSD, self-dual and LCD) codes over the non-unital non-local ring *H* were introduced. First, symplectic QSD and self-dual *H*-codes were characterized. It was shown that the minimum symplectic distance of an *H*-code can be determined from its associated binary codes. Then symplectic LCD *H*-codes were characterized. It was shown that the image under the map $\psi_H$ of any symplectic LCD *H*-code is trace self-orthogonal over $\mathbb{F}_4$. Finally, an algorithm to classify symplectic QSD *H*-codes was presented. This was used to classify such codes for small lengths. As future work, a similar study for other finite non-unitary rings with odd characteristic as given in [12, 28] is recommended.

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Adel Alahmadi, Research Group of Algebraic Structures and Applications, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah City 21589, Saudi Arabia

e-mail: analahmadi@kau.edu.sa

Widyan Bassafar, Research Group of Algebraic Structures and Applications, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah City 21589, Saudi Arabia

e-mail: whbasaffar@kau.edu.sa

Rowena Alma Betty, Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines

e-mail: rabetty@math.upd.edu.ph

Lucky Galvez, Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines

e-mail: legalvez@math.upd.edu.ph

Patrick Solé, I2M(CNRS, University of Aix-Marseille), 13009 Marseilles, France

e-mail: patrick.sole@telecom-paris.fr