Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products

1.   Introduction

Historically, the classification of self-dual codes over unitary rings and finite fields has rested on two pillars: an algorithm to generate short length codes, and a mass formula to signal the completion of the classification ^[7,15]. While the pionneers (Conway, Pless Sloane, and so on), were using various ad hoc methods like glueing theory to generate short codes ^[17], in recent years the build up method emerged as a systematic generation method ^[10,13,14]. By using a recursion on generator matrices, from a self-dual code of length $n$, it creates a self-dual code of length $n+h$ (with $h$ small and fixed). This is sometimes called a propagation rule of order $h.$ For concreteness, one may take $h = 2$ for binary codes ^[6] and codes over a certain ring of order $4$ ^[12], and $h = 4$ for ternary codes ^[13]. In ^[5], this technique was applied to a commutative non-unital ring with success.

In this paper, we initiate the study of self-dual codes over the ring $\mathit{E}_3,$ defined on two generators $a, b$ by the relations

This ring is a non-unital, non commutative ring of order $9.$ This notation is consistent with the classification of rings of order $p^2$ for $p$, a prime of ^[9]. While the use of finite fields as alphabets in Coding Theory dates back to its inception in the 1940's, and the use of finite rings from the 1980's, it is only in recent years that non-unitary rings have been used as alphabets at the cost of theoretical hurdles ^[1,2,3,4].

We will use a definition of self-dual codes for non-commutative rings introduced in ^[2]. This new notion of self-dual codes coincides with that of $QSD$ codes for that special ring. Also of interest are left self-dual ($LSD$) and right self-dual ($RSD$) codes, in agreement with the non-commutativity of the alphabet ring. In particular, we modify the propagation rules of ^[1] to produce self-orthogonal and self-dual codes, as well as one-sided self-dual codes. As an application, we classify the three types of codes considered in lengths at most $7.$ We derive mass formulas for the these three types that guarantee that the classification is complete.

The material is arranged as follows. The next section contains the preliminary notions and notations needed in the later sections. Section 3 derives building-up constructions for the three classes of codes mentioned. Section 4 applies these propagation rules to concrete classifications in short lengths. Section 5 concludes the article.

2.   Preliminaries

Let $\mathbb{F}_3^n$ represent the vector space of $n$-tuples over the $3$-element field $\mathbb{F}_3$. A ternary linear code $\mathcal{C}$ of length $n$ and dimension $k$, denoted shortly as an $\left[n, k \right]_3$ code, is a $k$-dimensional subspace of $\mathbb{F}_3^n$.

The number of nonzero coordinates of a vector ${\mathbf x} \in \mathbb{F}_3^n$ is called its Hamming weight $wt({\mathbf x})$. The Hamming distance $d({\mathbf x}, {\mathbf y})$ between two vectors ${\mathbf x}, {\mathbf y} \in \mathbb{F}_3^n$ is defined by $d({\mathbf x}, {\mathbf y}) = wt({\mathbf x}-{\mathbf y})$. The minimum distance of a linear code $\mathcal{C}$ is

A $3$-ary linear code of length $n$, dimension $k$, and minimum distance $d$ is said to be an $\left[n, k, d \right]_3$ code.

2.1.   Ring theory

Following ^[9], we define the ring $\mathit{E}_3$ of order 9 on two generators $a$ and $b$ by the relations:

Thus, $\mathit{E}_3$ has characteristic three, and consists of nine elements

where

These definitions immediately lead to the addition and the multiplication tables given as follow.

We may deduce from the multiplication table that this ring is non-commutative without identity and has unique maximal ideal $J_3 = \left\{ 0, f, g = 2f \right\}$ with residue field $\mathit{E}_3/J_3 \simeq \mathbb{F}_3.$

As a result, we have the following $f$-adic decomposition. It can be checked by inspection that any element ${\mathbf i}\in E_3$ can be expressed as ${\mathbf i} = a{\mathbf x}+f{\mathbf y},$ for unique scalars ${\mathbf x}, {\mathbf y} \in \mathbb{F}_3.$ We have defined a natural action of $\mathbb{F}_3$ on the ring $\mathit{E}_3$ by the rule

Note that, for all ${\mathbf r} \in \mathit{E}_3$, ${\mathbf x}, {\mathbf y} \in \mathbb{F}_3$, this action is "distributive" in the sense that ${\mathbf r} \left({\mathbf x} \oplus {\mathbf y} \right) = {\mathbf r} {\mathbf x}+{\mathbf r}{\mathbf y}$, where $\oplus$ denote the addition in $\mathbb{F}_3.$ When ${\mathbf x} \in \mathbb{F}_3^n$, and ${\mathbf r}\in \mathit{E}_3^n$, we will occasionally use the inner product notation $({\mathbf x}, {\mathbf r})$ to indicate

We define the reduction map modulo $S$ as $\pi :\mathit{E}_3\longrightarrow \mathit{E}_3/J_3 \simeq \mathbb{F}_3$ by

This map is extended in the natural way in a map from $\mathit{E}_3^n$ to $\mathbb{F}_3^n.$

2.2.   Codes over $\mathit{E}_3$

A linear $\mathit{E}_3$-code $\mathcal{C}$ of length $n$ is a one-sided $\mathit{E}_3$-submodule of $\mathit{E}_3^n$. It may be thought of as the $\mathit{E}_3$-span of the rows of a matrix called a generator matrix (we assume that these rows belong to $\mathcal{C}$). There are two ternary codes of length $n$ associated with the code $\mathcal{C}$. The residue code $\mathrm{res(\mathcal{C})}$ is just $\pi\left(\mathcal{C} \right)$, and the torsion code $\mathrm{tor(\mathcal{C})}$ is $\left\{ {\mathbf x} \in \mathbb{F}_3^n | \ f {\mathbf x} \in \mathcal{C}\right\}.$ It is easy to verify that $\mathrm{res(\mathcal{C})} \subseteq \mathrm{tor(\mathcal{C})}$ ^[3]. We denote the dimension of the residue code by $k_1$, and the dimension of the torsion code by $k_1 + k_2.$ Such a code $\mathcal{C}$ is of type $\{k_1, k_2\}$. A straightforward application of the first isomorphism theorem ^[3] shows that

By a result similar to [3, Theorem 1], we can show that every code $\mathcal{C}$ over $\mathit{E}_3$ of length $n$ and type $\left\{ k_1, k_2 \right\}$ is equivalent to a code with a generator matrix

where $Y$ is a matrix with entries in $\mathit{E}_3$, $X$ and $Z$ are matrices with entries from $\mathbb{F}_3$, and $I_{k_1}$, $I_{k_2}$ are identity matrices.

In fact, generator matrices of $\mathrm{res(\mathcal{C})}$ and $\mathrm{tor(\mathcal{C})}$ are given by

respectively.

Define the inner product of ${\mathbf x}$ and ${\mathbf y}$ in $\mathit{E}_3^{n}$ as $({\mathbf x}, {\mathbf y}) = x_1y_1+x_2y_2+\cdots +x_ny_n,$ where ${\mathbf x} = (x_1, x_2, \ldots, x_n)$ and ${\mathbf y} = (y_1, y_2, \ldots, y_n).$

The left dual code $\mathcal{C}^{\bot_L}$ of a code $\mathcal{C}$ is the module defined as

The right dual code $\mathcal{C}^{\bot_R}$ of a code $\mathcal{C}$ is the module defined as:

Thus, the left (respectively, right) dual of a left (respectively, right) module is a left (respectively, right) module.

A left self-dual code $\mathcal{C}$ satisfies $\mathcal{C} = \mathcal{C}^{\bot_L}$. Likewise, a right self-dual code $\mathcal{C}$ satisfies $\mathcal{C} = \mathcal{C}^{\bot_R}$.

The two-sided dual of $\mathcal{C}$, denoted by $\mathcal{C}^{\bot}$, is given by $\mathcal{C}^{\bot} = \mathcal{C}^{\bot _L}\cap \mathcal{C}^{\bot _R}.$ We say that a code $\mathcal{C}$ is self-dual $(SD)$ iff $\mathcal{C} = \mathcal{C}^{\bot _L}\cap \mathcal{C}^{\bot _R}.$ A code $\mathcal{C}$ is self-orthogonal $(SO)$ if for all $x, y \in \mathcal{C},$ $({\mathbf x}, {\mathbf y}) = 0.$

Clearly, $\mathcal{C}$ is a self-orthogonal code $\mathcal{C}$, iff $\mathcal{C}\subseteq \mathcal{C}^{\bot _L}\cap \mathcal{C}^{\bot _R}$.

Two $\mathit{E}_3$-codes are monomially equivalent if there is a monomial transformation of coordinates that maps one to the other. Here a monomial transformation is a matrix with entries in $\mathbb{F}_3$ and with exactly one element per row and per column. The parameters $(n, 3^k, d)$ of an $\mathit{E}_3$-code are identified with that of its image by $\phi,$ defined in the next subsection.

2.3.   Codes over $\mathbb{F}_9$

An additive code $\mathcal{C}$ of length $n$ over $\mathbb{F}_9$ is an additive subgroup of $\mathbb{F}_9^n.$ Thus, $\mathcal{C}$ contains $3^k$ codewords for some integer $0\le k\le 2n$, and is called an $(n, 3^k)$ code. If, furthermore, $\mathcal{C}$ has minimum distance $d,$ we write the parameters of $\mathcal{C}$ as $(n, 3^k, d).$ An additive code $\mathcal{C}$ over $\mathbb{F}_9$ can be represented by a $k\times n$ generator matrix with entries from $\mathbb{F}_9$ whose rows span $\mathcal{C}$, called a generator matrix. That is, $\mathcal{C}$ is the $\mathbb{F}_3$-span of its rows.

Let $\omega \in \mathbb{F}_9$ be such that $\omega^2 = \omega +1.$ Since the polynomial $t^2-t-1$ is irreducible over $\mathbb{F}_3$, we can write $\mathbb{F}_9 = \mathbb{F}_3\left[\omega \right].$ The trace map, $Tr:\mathbb{F}_9\longrightarrow \mathbb{F}_3$, is defined as $Tr(x) = x+x^3$.

Every linear $\mathit{E}_3$-code $\mathcal{C}$ is attached with an additive $\mathbb{F}_9$-code $\phi(\mathcal{C})$ by the alphabet substitution

extended naturally to $\mathbb{F}_9^n$. The parameters $(n, 3^k, d)$ of an $\mathit{E}_3$-code are identified with that of its image under $\phi.$ It can be checked that, for all ${\mathbf x} \in \mathit{E}_3^n$, we have $Tr(\phi({\mathbf x})) = \pi({\mathbf x})$, and thus $\mathrm{res(\mathcal{C})} = Tr(\phi(\mathcal{C}))$.

Similarly, we see that $\mathrm{tor(\mathcal{C})}$ is the so-called subfield subcode of $\phi(\mathcal{C})$, that is $\mathbb{F}_3^n \cap \phi(\mathcal{C})$.

2.4.   Weight enumerators

We recall from ^[17] that the weight enumerator of any linear or additive code $\mathcal{C}$ of length $n$ is the polynomial

where the sequence $A_1, ......, A_n$ is the weight distribution of $\mathcal{C}$. That is, $A_i$ is the number of codewords in $\mathcal{C}$ of weight $i$.

Lemma 1. If $\mathcal{C}$ is a linear code of length $n$ over $\mathit{E}_3$ with weight enumerator $\mathrm{W}_\mathcal{C}({\mathbf x}, {\mathbf y})$, then the weight enumerator of the dual code $\mathcal{C}^\bot$ is given by

In particular, the weight enumerator of a self-dual code is invariant under the matrix group generated by $\frac{1}{3}\begin{pmatrix} 1 & 8 \\ 1 & -1 \end{pmatrix}.$

Proof. The first statement, analogous to the MacWilliams identity for linear codes over $\mathbb{F}_q$, follows from the general theory [17, Theorem 13], since our trace inner product $\phi(\mathcal{C})$ is an $(n, 3^{2k_1+k_2})$ additive code over $\mathbb{F}_9$ with

The second statement follows from the first, by noticing that, for a self-dual code, we have $\mathrm{W}_{\mathcal{C}^\bot } = \mathrm{W}_{\mathcal{C} }.$ □

2.5.   Mass formulas

Let $\varphi_{n, k}$ denote the number of distinct self-orthogonal ternary codes having parameters $[n, k_1]$ (see [15, Theorem 4.7]. We define the Gaussian coefficient $\begin{pmatrix} k \\ r \end{pmatrix}_3$ for $r\le k$ as

which gives the number of subspaces of dimension $r$ contained in a $k$-dimensional vector space over $\mathbb{F}_3$.

In what follows, we present a mass formula for self-orthogonal codes over $\mathit{E}_3$ which is a characteristic 3-version of [4, Theorem 8].

Theorem 1. For all lengths $n,$ and for the type $\left\{ k_1, k_2 \right\}$, where $k_1 \ge 1,$ the number of self-orthogonal codes over $\mathit{E}_3$ is

If $k_1 = 0,$ then

The following mass formulas follow by the usual counting technique under group action. Corollaries 1–4 are a consequence of Theorem 1.

Corollary 1. For given length $n$ and type $\left\{ k_1, k_2 \right\}$, with $0\le k_1, k_2\le n$, we have

where $\mathcal{C}$ runs over distinct representatives of equivalence classes under monomial column permutations of $SO$ codes of length n and type $(k_1, k_2)$.

Corollary 2. For given length $n$ and type $\left\{ k_1, k_2 \right\}$, with $k_1 \ge 1$, we have

where $\mathcal{C}$ runs over distinct representatives of equivalence classes under monomial column permutations of $SD$ codes of length $n$ and type $\left\{ k_1, k_2 \right\}$.

Corollary 3. For given length $n$ and type $\left\{0, k_2 \right\}$, with $k_2 \ge 1$, we have

where $\mathcal{C}$ runs over distinct representatives of equivalence classes under monomial column permutations of $RSD$ codes of length $n$ and type $\left\{0, k_2 \right\}$.

Corollary 4. For given length $n$ and type $\left\{ \frac{n}{2}, 0 \right\}$, with $\frac{n}{2} \ge 1$, we have

where $\mathcal{C}$ runs over distinct representatives of equivalence classes under monomial column permutations of $LSD$ codes of length $n$ and type $\left\{ \frac{n}{2}, 0 \right\}$.

3.   Construction

In this section we provide the building-up construction for self-orthogonal codes, (one-sided) self-dual, and self-dual codes over $\mathit{E}_3$. Our building-up construction needs the following theorems.

Theorem 2. Let $\mathcal{C}$ be an $\mathit{E}_3$ code of length $n$, with residue code $\mathcal{C}_1$ and torsion code $\mathcal{C}_2.$ The following hold:

● If $\mathcal{C}$ is a linear code of length $n$ over $\mathit{E}_3$, then $\mathcal{C} = a \mathcal{C}_1 + f \mathcal{C}_2.$

● If $\mathcal{C}_1$ is self-orthogonal with $\mathcal{C}_1\subseteq \mathcal{C}_2\subseteq \mathcal{C}_1^\bot$, then $\mathcal{C}$ is a self-orthogonal code.

● If, furthermore, $\left| \mathcal{C} \right| = 3^n$, then $\mathcal{C}_2 = \mathcal{C}_1^\bot$.

Proof. $\underline{{\text{The first statement:}}}$ It suffices to prove that $\mathcal{C} \subseteq a \mathcal{C}_1 + f \mathcal{C}_2$.

Let ${\mathbf i} \in \mathcal{C}$. We can write ${\mathbf i}$ in $f$-adic decomposition form as ${\mathbf i} = a{\mathbf x}+f{\mathbf y}$ where ${\mathbf x}, {\mathbf y} \in \mathbb{F}_3^n$.

Since $\pi(a{\mathbf x}+f{\mathbf y}) = {\mathbf x}$, it follows that ${\mathbf x} \in \mathrm{res(\mathcal{C})}$.

By following [3, Lemma 3], we have $a{\mathbf x} \in \mathcal{C}$. Also by a linearity of $\mathcal{C}$, we can see that $f{\mathbf y} \in \mathcal{C}$, and hence ${\mathbf y} \in \mathrm{tor(\mathcal{C})}$.

This proves that $\mathcal{C} \subseteq a \mathcal{C}_1 + f \mathcal{C}_2$.

$\underline{{\text{The second statement:}}}$ Suppose ${\mathbf i}_1, {\mathbf i}_2 \in \mathcal{C}$. We can write ${\mathbf i}_1$ and ${\mathbf i}_2$ in $f$-adic decomposition form as ${\mathbf i}_1 = a{\mathbf x}_1+f{\mathbf y}_1$ and ${\mathbf i}_2 = a{\mathbf x}_2+f{\mathbf y}_2$ where ${\mathbf x}_1, {\mathbf x}_2 \in \mathrm{res(\mathcal{C})}$ and ${\mathbf y}_1, {\mathbf y}_2 \in \mathrm{tor(\mathcal{C})}$. Compute $({\mathbf i}_1, {\mathbf i}_2)$,

Since $\mathrm{tor(\mathcal{C})} \subseteq \mathrm{res(\mathcal{C})}^\bot$, it follows that $({\mathbf x}_1, {\mathbf x}_2) = ({\mathbf x}_2, {\mathbf y}_1) = 0$, which is self-orthogonal. Thus, $\mathcal{C}$ is self-orthogonal.

$\underline{{\text{The last statement:}}}$ It follows that $\mathcal{C}_2 = 3^{n-k_1} = \mathcal{C}_1^\bot$. □

The next result control the linear structure of SO codes.

Corollary 5. For any $SO$ codes $\mathcal{C}$ over $\mathit{E}_3$ of length $n$, we have

(i) $\mathrm{tor(\mathcal{C})} \subseteq \mathrm{res(\mathcal{C})}^\bot.$

(ii) $\mathrm{res(\mathcal{C})}$ must be a self-orthogonal ternary code.

Proof. For ($i$), let ${\mathbf y} \in \mathrm{tor(\mathcal{C})}$, then $f {\mathbf y} \in \mathcal{C}$ and ${\mathbf q} \in \mathcal{C}$, we can write ${\mathbf q}$ in $f$-adic decomposition form as ${\mathbf q} = a{\mathbf u} + f{\mathbf v}$, where ${\mathbf u} \in \mathrm{res(\mathcal{C})}$ and ${\mathbf v} \in \mathrm{tor(\mathcal{C})}$.

Thus, ${\mathbf y} \in \mathrm{res(\mathcal{C})}^\bot$ and $\mathrm{tor(\mathcal{C}}) \subseteq \mathrm{res(\mathcal{C})}^\bot$.

For ($ii$), By ($i$), and the fact that $\mathrm{res(\mathcal{C})}\subseteq \mathrm{tor(\mathcal{C})}$, it follows that

□ The following theorem characterizes one-sided self-dual codes over $\mathit{E}_3.$

Theorem 3. For any linear code $\mathcal{C}$ over $\mathit{E}_3$, we have

(i) $\mathcal{C}^{\bot_L} = a \ \mathrm{res(\mathcal{C})}^\bot + f \ \mathrm{res(\mathcal{C})}^\bot$.

(ii) $\mathcal{C}^{\bot_R} = a \ \mathrm{tor(\mathcal{C})}^\bot +f \ \mathbb{F}_3^n.$

Proof. For ($i$), we need to prove that $\mathrm{res(\mathcal{C}^{\bot _L})} = \mathrm{res(\mathcal{C})}^\bot = \mathrm{tor(\mathcal{C}^{\bot _L})}$.

From the fact $\mathrm{res(\mathcal{C}^{\bot _L})} \subseteq \mathrm{tor(\mathcal{C}^{\bot _L})}$, it follows that $\mathrm{res(\mathcal{C}^{\bot _L})} \subseteq \mathrm{res(\mathcal{C})}^\bot \subseteq \mathrm{tor(\mathcal{C}^{\bot _L})}.$

Now, let ${\mathbf y} \in \mathrm{tor(\mathcal{C}^{\bot _L})}$, then $f {\mathbf y} \in \mathcal{C}^{\bot _L}$ and ${\mathbf x} \in \mathrm{res(\mathcal{C})}$, then $a{\mathbf x} \in \mathcal{C}.$ By definition of $\mathcal{C}^{\bot _L}$, we can see that

Thus, ${\mathbf y} \in \mathrm{res(\mathcal{C})}^\bot$ and $\mathrm{tor(\mathcal{C}}^{\bot _L}) \subseteq \mathrm{res(\mathcal{C})}^\bot$.

On the other hand, given ${\mathbf r} \in \mathrm{res(\mathcal{C})}^\bot$ and ${\mathbf i} \in \mathcal{C}$, ${\mathbf i}$ may be expressed in $f$-adic decomposition form as ${\mathbf i} = a{\mathbf u}+f{\mathbf v}$, where ${\mathbf u} \in \mathrm{res(\mathcal{C})}$ and ${\mathbf v} \in \mathrm{tor(\mathcal{C})}, respectively.$ It can be seen that

by definition of $\mathcal{C}^{\bot}$. Thus, $a{\mathbf r} \in \mathcal{C}^{\bot_L}$, and ${\mathbf r} \in \mathrm{res(\mathcal{C}}^{\bot_L}).$ This implies that, $\mathrm{res(\mathcal{C})}^\bot \subseteq \mathrm{res(\mathcal{C}}^{\bot_L})$. Therefore,

For ($ii$), suppose ${\mathbf y} \in \mathrm{tor(\mathcal{C})}^\bot \subseteq \mathrm{res(\mathcal{C})}^\bot$, and ${\mathbf i} \in \mathcal{C}$, we can write ${\mathbf i}$ in $f$-adic decomposition form as ${\mathbf i} = a{\mathbf u}+f{\mathbf v}$, where ${\mathbf u} \in \mathrm{res(\mathcal{C})}$ and ${\mathbf v} \in \mathrm{tor(\mathcal{C})}$. By definition of $\mathcal{C}^{\bot}$, we have

Thus, $a{\mathbf y} \in \mathcal{C}^{\bot _R}$, and ${\mathbf y} \in \mathrm{res(\mathcal{C}}^{\bot _R})$. This implies that, $\mathrm{tor(\mathcal{C})}^\bot \subseteq \mathrm{res(\mathcal{C}}^{\bot _R}).$ Conversely, let ${\mathbf x} \in \mathrm{res(\mathcal{C}}^{\bot _R})$, then $a{\mathbf x} \in \mathcal{C}^{\bot _R}$. Let ${\mathbf y} \in \mathrm{tor(\mathcal{C}})$, then $f{\mathbf y} \in \mathcal{C}$. Observe that

Thus, $f{\mathbf y} \in \mathcal{C}^\bot$ and ${\mathbf y} \in \mathrm{tor(\mathcal{C}})^{\bot }$. This implies that $\mathrm{res(\mathcal{C}}^{\bot _R}) = \mathrm{tor(\mathcal{C}})^{\bot}.$

Now to prove $\mathrm{tor(\mathcal{C}}^{\bot _R}) = \mathbb{F}_3^n$, it suffices to show that $\mathbb{F}_3^n \subseteq \mathrm{tor(\mathcal{C}}^{\bot _R}).$

Let ${\mathbf r} \in \mathbb{F}_3^n$, and ${\mathbf i} \in \mathcal{C}$, we can write ${\mathbf i}$ in $f$-adic decomposition form as ${\mathbf i} = a{\mathbf u}+f{\mathbf v}$, where ${\mathbf u} \in \mathrm{res(\mathcal{C})}$ and ${\mathbf v} \in \mathrm{tor(\mathcal{C})}$. We can see that

Thus, $f{\mathbf r} \in \mathcal{C}^{\bot _R}$, then ${\mathbf r} \in \mathrm{tor(\mathcal{C}}^{\bot _R})$, and so $\mathbb{F}_3^n \subseteq \mathrm{tor(\mathcal{C}}^{\bot _R}).$ This completes the proof. □

Corollary 6. For any linear code $\mathcal{C}$ over $\mathit{E}_3$, we have

(i) $\mathcal{C}$ is left self-dual $(LSD)$ iff $\mathcal{C}$ is of type $\left\{ \frac{n}{2}, 0 \right\}$.

(ii) $\mathcal{C}$ is right self-dual $(RSD)$ iff $\mathcal{C} = f \ \mathbb{F}_3^n$.

The lengths of LSD codes can be determined completely.

Corollary 7. Left self-dual codes over $\mathit{E}_3$ of length $n$ occurs if and only if $n$ is a multiple of $4$.

Proof. From Theorem 3, $\mathcal{C}$ is left self-dual iff $\mathrm{res(\mathcal{C})}$ is a self-dual ternary code. Then, by [16, Theorem 3], a self-dual code over $\mathbb{F}_3$ of length $n$ exists only if $n$ is a multiple of $4$.

This completes the proof. □ The next theorem characterizes $SD$ codes.

Theorem 4. Assume $\mathcal{C}$ is an $\mathit{E}_3$-code of length $n$, $\mathcal{C}$ is a $SD$ code if and only if

moreover, $\left| \mathcal{C} \right| = 3^n$.

Proof. We know that $\mathcal{C}$ is self-dual code over $\mathit{E}_3$ iff $\mathcal{C} = \mathcal{C}^{\bot} = \mathcal{C}^{\bot _R} \cap \mathcal{C}^{\bot _L}$. By Theorem 2, we can see that

Thus, by Theorem 3, it can be seen that

□

Corollary 8. For any linear code $\mathcal{C}$ over $\mathit{E}_3$, we have:

(1) If $\mathcal{C}$ is $SD$ with $k_1 = 0$, then $\mathcal{C}$ is $RSD$.

(2) If $\mathcal{C}$ is $SD$ with $k_2 = 0$, then $\mathcal{C}$ is $LSD$.

The next result bounds the minimum distances of SD and LSD codes.

Corollary 9. (i) The minimum distance $d(\mathcal{C})$ of a self-dual code over $\mathit{E}_3$ is less than or equal to $\min \left\{d(\mathrm{res(\mathcal{C})}), d(\mathrm{res(\mathcal{C})}^\bot)\right\}$.

(ii) If $\mathcal{C}$ is a left self-dual code $\mathcal{C}$ over $\mathit{E}_3$, then the minimum distance $d(\mathcal{C})$ is equal to $d(\mathrm{res(\mathcal{C})})$.

Proof. For ($i$), by Theorem 4, we have $\mathcal{C} = a \ \mathrm{res(\mathcal{C})} + f \ \mathrm{res(\mathcal{C})}^\bot.$

Let $d_1$ and $d_2$ denote the minimum distances of $\mathrm{res(\mathcal{C})}$ and $\mathrm{res(\mathcal{C})}^\bot$, respectively.

By definition of $\mathrm{tor(\mathcal{C})}$, we have $d(\mathcal{C})\le d_2$. By [3, Lemma 3], we have $a \ \mathrm{res(\mathcal{C})}\subseteq \mathcal{C}$, which indicates that $d(\mathcal{C})\le d_1$. It follows that $d(\mathcal{C})\le min\left\{ d_1, d_2 \right\}$.

For ($ii$), by Theorem 3, we have $\mathcal{C} = a \ \mathrm{res(\mathcal{C})}^\bot + f \ \mathrm{res(\mathcal{C})}^\bot,$ where $\mathrm{res(\mathcal{C})}^\bot = \mathrm{res(\mathcal{C})}.$ This means that $d_1 = d_2.$

We now establish that $d(\mathcal{C}) \ge d_1$. Assume ${\mathbf q}$ is in $\mathcal{C}$ and $wt({\mathbf q}) = d$. Because ${\mathbf r} \in \mathrm{res(\mathcal{C})}^\bot$ and ${\mathbf s} \in \mathrm{res(\mathcal{C})}^\bot$, ${\mathbf q} = a{\mathbf r}+f{\mathbf s}$ by Theorem 2. We have the following three scenarios, which rely on ${\mathbf r}$ and ${\mathbf s},$ since $\mathcal{C}$ is nonzero:

$(1)$ $wt({\mathbf q}) = wt(a {\mathbf r}) = wt({\mathbf r}) \, \ \text{if} \, \ {\mathbf r}\neq 0 \, \ \text{and} \, \ {\mathbf s} = 0$,

$(2)$ $wt({\mathbf q}) = wt(f {\mathbf s}) = wt({\mathbf s}) \, \ \text{if} \, \ {\mathbf r} = 0 \, \ \text{and} \, \ {\mathbf s}\neq 0$,

$(3)$ $wt({\mathbf q}) = wt(a {\mathbf r}+f {\mathbf s})\ge wt({\mathbf ar}) = wt({\mathbf r}) \, \ \text{if} \, \ {\mathbf r}, {\mathbf s}\neq 0.$

It follows that $d(\mathcal{C}) \ge d_1$ since $d_1 = d_2$. Thus, from $(i)$, we conclude that $d(\mathcal{C}) = d_1.$ □

3.1.   Construction of self-orthogonal codes

In this subsection, we present the build-up construction method for self-orthogonal codes over $\mathit{E}_3$. The following theorem is a propagation rule of order three which increases the number of generators by one.

Theorem 5. Let $\mathcal{C}_{0}$ be a self-orthogonal code over $\mathit{E}_3$ of length $n$ with generator matrix $\mathcal{G}_{0} = ({\mathbf r}_{i})$ where ${\mathbf r}_{i}$ is the $i$th row of $\mathcal{G}_{0}$, for $i = 1, 2, \ldots, m$. Let ${\mathbf x}\in\mathbb{F}_{3}^{n}$ and $\alpha, \beta, \gamma \in \mathit{E}_3,$ such that $\alpha+\beta+\gamma = 0$. Then the code $\mathcal{C}$ with the following generator matrix

is a self-orthogonal code of length $n+3$ if:

(i) $({\mathbf x}, {\mathbf x}) = 1$, and $\alpha, \beta, \gamma \in \mathit{E}_3 \setminus J_3$; or

(ii) $({\mathbf x}, {\mathbf x}) = -1$ and $(\alpha \neq 0, \, \ \beta = 0, \, \ \gamma = 2\alpha)$.

Proof. It suffices to show that the rows of $\mathcal{G}$ are orthogonal to each other. Let ${\mathbf y}_{0} = \left(\begin{array}{cccc} \alpha & \beta & 0 & \gamma{\mathbf x}\end{array}\right)$, the first row of $\mathcal{G}$ and ${\mathbf y}_{i} = \left(\begin{array}{cccc} ({\mathbf x}, {\mathbf r}_{i}) & ({\mathbf x}, {\mathbf r}_{i}) & ({\mathbf x}, {\mathbf r}_{i}) & {\mathbf r}_{i}\end{array}\right)$, the $i+1$st row of $\mathcal{G}$, for $i = 1, 2, \ldots, m$.

$(i)$ If $({\mathbf x}, {\mathbf x}) = 1$ and $\alpha, \beta, \gamma \in \mathit{E}_3 \setminus J_3$, then

for all $i, j = 1, 2, \ldots, m$.

$(ii)$ If $({\mathbf x}, {\mathbf x}) = -1$ and $(\alpha \neq 0, \, \ \beta = 0, \, \ \gamma = 2\alpha)$, then

for all $i, j = 1, 2, \ldots, m$.

In both cases, we see that ${ \mathcal{C}}$ is a self-orthogonal code. □

Example 1. We construct self-orthogonal codes of length $6$ derived from length 3. By Theorem 2, we have the self-orthogonal $(3, 3^2, 3)$ code with generator matrix $\mathcal{G}_0$ given by

Using Theorem 5 from $\mathcal{G}_0$, we get three non-equivalent monomial $SO$ $(6, 3^4, 3)$ codes, with generator matrices:

Their Hamming weight enumerators are

and

respectively.

Remark 1. The build-up construction method given in Theorem 5, cannot be also used for self-dual codes as the cardinality of the code $\mathcal{C}$ is not $3^{n+3}.$

Remark 2. If any two self-orthogonal $\mathit{E}_3$-codes are monomially equivalent, then their residue and torsion codes are also monomially equivalent, but the converse is not necessarily true. For example, when $n = 3$ and type $\left\{ 1, 0 \right\}$, the codes $\mathcal{C}_1$ and ${\mathcal{C}}_2$ with generator matrices $\begin{pmatrix} a & a & a\\ \end{pmatrix}$ and $\begin{pmatrix} a & b & h\\ \end{pmatrix}$, respectively. They have the same residue and torsion codes, but are not monomially equivalent.

3.2.   Construction of one-sided self-dual codes

We will provide a construction approach for one-sided self-dual codes over $\mathit{E}_3$ in this subsection. Initially, this structure was provided for left self-dual codes with two extra generators and a length increase of four. We only need to take into consideration the situation in which the length $n$ is a multiple of $4$, as a result of Corollary 7.

Theorem 6. Let $\mathcal{C}_{0}$ be a left self-dual code over $\mathit{E}_3$ of length $n$ with generator matrix $\mathcal{G}_{0} = ({\mathbf r}_{i})$ where ${\mathbf r}_{i}$ is the $i$th row of $\mathcal{G}_{0}$, for $i = 1, 2, \ldots, m$. Let ${\mathbf x}_1, {\mathbf x}_2 \in \mathbb{F}_{3}^{n}$ such that $({\mathbf x}_1, {\mathbf x}_2) = 0$, and $({\mathbf x}_i, {\mathbf x}_i) = 2$ for $i = 1, 2$. For $1 \le i \le m$, define $u_i = ({\mathbf x}_1, {\mathbf r}_i)$ and $v_i = ({\mathbf x}_2, {\mathbf r}_i)$. If $\alpha \in \mathit{E}_3 \setminus J_3$, then the code $\mathcal{C}$ with the following generator matrix

is a left self-dual code of length $n+4$.

Proof. Let ${\mathbf y}_0 = \left(\begin{array}{ccccc} \alpha & 0 & 0 & 0 & 2\alpha {\mathbf x}_1 \end{array} \right)$ and ${\mathbf y}_0' = \left(\begin{array}{ccccc} 0 & \alpha & 0 & 0 & 2\alpha {\mathbf x}_2 \end{array} \right)$. Then

Now, for $1 \leq i \leq m$, let ${\mathbf y}_i = \left(\begin{array}{ccccc} u_i & v_i & u_i+v_i & 2u_i+v_i & {\mathbf r}_i \end{array} \right)$. So, for $1 \leq j \leq m$,

Thus, $\mathcal{C}$ is self-orthogonal and since $\left| \mathcal{C} \right| = 9^2\left| \mathcal{C}_0 \right| = 3^{n+4}$, $\mathcal{C}$ is left self-dual. □

Define $\widehat{\mathcal{C}_0}$ to be the span of the last $m$ rows of $\mathcal{G}$, and for every $y \in \mathit{E}_3$ write

The construction in Theorem 6 maybe demonstrated to be equivalent to

Example 2. We construct a left self-dual code of length $8$ derived from length 4. By Theorem 3, we have the left self-duall $(4, 3^4, 3)$ code with generator matrix $\mathcal{G}_1$ given by

Using Theorem 6 from $\mathcal{G}_1$ with $\alpha = a$, we get only one non-equivalent monomial left self-dual $(8, 3^8, 3)$ code with generator matrix

Its Hamming weight enumerator is

Repeating this process using $\mathcal{G}_{1, 1}$ with ${\mathbf x}_1 = (1, 1, 0, 0, 0, 0, 0, 0)$, we get only three non-equivalent monomial left self-dual $(12, 3^{12}, 3)$ codes with generator matrices

and

Their Hamming weight enumerators are

and

respectively.

Corollary 10. The minimum Hamming weight of left self-dual codes over $\mathit{E}_3$ of length $n$ is a multiple of $3$.

Proof. Following Corollary 9, the minimum distance $d(\mathcal{C})$ of a left self-dual code $\mathcal{C}$ is equal to $d(\mathrm{res(\mathcal{C})})$. Thus, by [11, Lemma 1], self-dual codes over $\mathbb{F}_3$ of length $n$ exist only if $n$ is a multiple of $4$ and a minimum weight is a multiple of $3$. □

In the following theorem, we will now describe the general approach that we will utilize in finding construction methods for right self-dual codes over $\mathit{E}_3$.

Theorem 7. Let $\mathcal{C}_0$ be a right self-dual code of length $n$ over $\mathit{E}_3$, with generator matrix $\mathcal{G}_0 = ({\mathbf r}_i)$, where $r_i$ is the $i$-th row of $\mathcal{G}_0$, for $1 \le i\le m$. Then the code $\mathcal{C}$ with the following generator matrix

is a a right self-dual code of length $n+h$.

Proof. We can show that $\mathcal{G}$ generates a self-orthogonal code $\mathcal{C}$ in a similar way as given in Theorem 6.

Now, according to $\left| \mathcal{C} \right| = 3^h\left| \mathcal{C}_0 \right| = 3^{n+h}$, $\mathcal{C}$ is self-dual. □

Corollary 11. Every one-sided self-dual code is also self-dual.

Proof. Let $\mathcal{C}$ be a left self-dual in order to demonstrate that every left self-dual is self-dual. $\mathrm{res(\mathcal{C})} = \mathrm{res(\mathcal{C})^{\bot}}$ and $\mathrm{tor(\mathcal{C})} = \mathrm{res(\mathcal{C})^{\bot}}$ are the results of Theorem 3. Hence, $\mathcal{C}$ is a self-dual code according to Theorem 4.

To prove that every right self-dual is self-dual, let $\mathcal{C}$ be right self-dual. Also, by use Theorem 3, we see that $\mathcal{C} = a \left\{ 0 \right\}+ f \mathbb{F}_3^n.$ Observe that $\mathrm{tor(\mathcal{C})} = \mathrm{res(\mathcal{C})}^\bot$. Then, by Theorem 4, $\mathcal{C}$ is a self-dual code. □

Example 3. The linear code of length 4 with the two generators

is a self-dual and left self-dual, but not right self-dual.

Example 4. The linear code of length 3 defined by

is a self-dual and right self-dual code, but not left self-dual.

Remark 3. We note that a similar construction method in Theorems 6 and 7 can be applied to self-dual codes over $\mathit{E}_3$.

3.3.   Construction of self-dual codes

The following result constructs a self-dual code of length $n+3$ from a self-dual code of length $n$.

Theorem 8. Let $\mathcal{C}_{0}$ be a self-dual code over $\mathit{E}_3$ of length $n$ with generator matrix $\mathcal{G}_{0} = ({\mathbf r}_{i})$ where ${\mathbf r}_{i}$ is the $i$th row of $\mathcal{G}_{0}$, for $i = 1, 2, \ldots, m$. Let ${\mathbf x}\in\mathbb{F}_{3}^{n}$ and $\alpha, \beta, \gamma \in \mathit{E}_3$, such that $\alpha+\beta+\gamma = 0$. Let $\sigma\in J_3$, non-zero element. Then the code $\mathcal{C}$ with the following generator matrix

is a self-dual code of length $n+3$ if:

(1) $({\mathbf x}, {\mathbf x}) = 1$, and $(\alpha, \beta, \gamma \in \mathit{E}_3 \setminus J_3)$; or

(2) $({\mathbf x}, {\mathbf x}) = -1$, and $(\alpha = 0, \, \ \beta \in \mathit{E}_3 \setminus J_3, \, \ \gamma = 2\beta)$.

Proof. We first show that $\mathcal{C}$ is self-orthogonal. Let ${\mathbf y}_{0} = \left(\begin{array}{cccc} \alpha & \beta & 0 & \gamma{\mathbf x}\end{array}\right)$, the first row of $\mathcal{G}$, and ${\mathbf y}_{0}' = \left(\begin{array}{cccc} 0 & \sigma & 0 & 2\sigma {\mathbf x}\end{array}\right)$, the second row of $\mathcal{G}$. For $1 \le i \le m$, let ${\mathbf y}_{i} = \left(\begin{array}{cccc} ({\mathbf x}, {\mathbf r}_{i}) & ({\mathbf x}, {\mathbf r}_{i}) & ({\mathbf x}, {\mathbf r}_{i}) & {\mathbf r}_{i}\end{array}\right)$. Then,

$(i)$ If $({\mathbf x}, {\mathbf x}) = 1$ and $(\alpha, \beta, \gamma \in \mathit{E}_3 \setminus J_3)$, then

for all $i, j = 1, 2, \ldots, m$.

$(ii)$ If $({\mathbf x}, {\mathbf x}) = -1$ and $(\alpha = 0, \, \ \beta \in \mathit{E}_3 \setminus J_3, \, \ \gamma = 2\beta)$, then

for all $i, j = 1, 2, \ldots, m$.

In both cases, we see that ${\mathcal{C}}$ is a self-orthogonal code. Since $|\mathcal{C}| = 9^13^1|\mathcal{C}_0| = 3^{n+3}$, $\mathcal{C}$ is a self-dual code of length $n+3$. □

Example 5. We construct self-dual codes of length 7 derived from length 4. By Theorem 4, we have the self-dual $(4, 3^4, 1)$ code with generator matrix $\mathcal{G}_3 = \begin{pmatrix} a & 0 & a & a\\ 0 &f & 0 & 0\\ 0 & 0 & f & g \\ \end{pmatrix}.$

Using Theorem 8 from $\mathcal{G}_3$, we get five non-equivalent monomial self-dual codes with generator matrices:

Their Hamming weight enumerators are

and

respectively.

Theorem 9. Two self-dual $\mathit{E}_3$-codes are monomially equivalent if and only if their residue codes are also equivalent.

Proof. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two monomially equivalent codes over $\mathit{E}_3$, then their residue codes are also equivalent as $a \ \mathrm{res(\mathcal{C}_1)} \subseteq \mathcal{C}_1$ and $a \ \mathrm{res(\mathcal{C}_2)} \subseteq \mathcal{C}_2$.

Conversely, Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two self-dual $\mathit{E}_3$-codes, and $\mathrm{res(\mathcal{C}_1)}$ and $\mathrm{res(\mathcal{C}_2)}$ monomially equivalent codes. Then, there is a monomial matrix $M$ sends a code $res(\mathcal{C}_{1})$ into the equivalent code such that

The set of all monomials such that $\mathrm{res(\mathcal{C}_2)} = \mathrm{res(\mathcal{C}_{1})}$ forms the automorphism group $G(\mathrm{res(\mathcal{C}_{1})})$ of the code $\mathrm{res(\mathcal{C}_{1})}$, and as $\mathrm{res(\mathcal{C}_2)^\bot} = \mathrm{res(\mathcal{C}_{1})^\bot} M.$

Thus, from Theorem 4, we have

By Eq (3.1), we have $\mathcal{C}_2 = \mathcal{C}_{1} M$, proving that $\mathcal{C}_1$ and $\mathcal{C}_2$ are monomially equivalent. □

Remark 4. Every one-sided self-dual code is also a self-dual code, but not conversely as the next example shows.

Example 6. The linear code with the three generators $\mathcal{G} = \begin{pmatrix} a & 0 & a & a\\ 0 & f & 0 & 0\\ 0 & 0 & f & g \end{pmatrix},$ is a self-dual code, but neither left- nor right-self-dual.

4.   Computational results

In this section, we use the multilevel constructs in Theorems 2–4 to categorize self-orthogonal, (one-sided) self-dual, and self-dual $\mathit{E}_3$-codes of length $n < 8$ with residue dimension $k_1 = 0, 1, 2$. All the computer calculations in this section were performed in Magma ^[8].

4.1.   Length 1

There is one right self-dual code over $\mathit{E}_3$ of type $\left\{ 0, 1 \right\}$, with generator matrix in Table 2.

4.2.   Length 2

There is one right self-dual code over $\mathit{E}_3$ of type $\left\{ 0, 2 \right\}$, with generator matrix in Table 3.

4.3.   Length 3

For type $\left\{ 0, 2 \right\}$, there are three distinct self-orthogonal codes over $\mathit{E}_3$, with generator matrices in Table 4.

For type $\left\{ 0, 3 \right\}$, there is one right self-dual code over $\mathit{E}_3$, with generator matrix in Table 5.

For type $\left\{ 1, 0 \right\}$, there are two distinct self-orthogonal codes over $\mathit{E}_3$, with generator matrices in Table 6.

For type $\left\{ 1, 1 \right\}$, there is one distinct self-dual code over $\mathit{E}_3$, with generator matrix in Table 7.

4.4.   Length 4

For type $\left\{ 0, 4 \right\}$, there is one right self-dual code over $\mathit{E}_3$, with generator matrix in Table 8.

For type $\left\{ 1, 0 \right\}$, there are four distinct self-orthogonal codes over $\mathit{E}_3$, with generator matrices in Table 9.

For type $\left\{ 1, 1 \right\}$, there are six distinct self-orthogonal codes over $\mathit{E}_3$, with generator matrices in Table 10.

For type $\left\{ 1, 2 \right\}$, there is only one distinct self-dual codes over $\mathit{E}_3$, with generator matrix in Table 11.

For type $\left\{ 2, 0 \right\}$, there is only one left self-dual code over $\mathit{E}_3$, with generator matrix in Table 12.

4.5.   Length 5

For type $\left\{ 0, 5 \right\}$, there is one right self-dual code over $\mathit{E}_3$, with generator matrix in Table 13.

For type $\left\{ 1, 0 \right\}$, there are six distinct self-orthogonal codes, with generator matrices in Table 14.

For type $\left\{ 1, 2 \right\}$, there are ten distinct self-orthogonal codes, with generator matrices in Table 15.

For type $\left\{ 1, 3 \right\}$, there is only one distinct self-dual codes over $\mathit{E}_3$, with generator matrix in Table 16.

4.6.   Length 6

For type $\left\{ 0, 6 \right\}$, there is one right self-dual code over $\mathit{E}_3$, with generator matrix in Table 17.

For type $\left\{ 1, 3 \right\}$, there are 13 distinct self-orthogonal codes, with generator matrices in Table 18.

For type $\left\{ 1, 4 \right\}$, there are two distinct self-dual codes over $\mathit{E}_3$, with generator matrices in Table 19.

4.7.   Length 7

For type $\left\{ 0, 7 \right\}$, there is one right self-dual code over $\mathit{E}_3$, with generator matrix in Table 20.

For type $\left\{ 1, 5 \right\}$, there are two distinct self-dual codes over $\mathit{E}_3$, with generator matrices in Table 21.

4.8.   Examples

According to the mass formula in subsection 2.5, we enumerate all equivalent codes obtained under the action of the monomial group, and illustrate how the mass formula can be used in the classification of $\mathit{E}_3$-codes, that is, we find representatives for the equivalence classes of $\mathit{E}_3$-codes for each length and type. The following examples illustrate our results.

Example 7. Let $\mathcal{C}$ be the set of self-orthogonal codes over $\mathit{E}_3$ of length 3 and type $\left\{ 1, 0 \right\}$ given by the following generator matrices:

The codes in $\mathcal{C}$ are inequivalent and they have an automorphism group of order 12 and 6 respectively. Therefore, from Corollary 1,

It demonstrates that, up to monomial equivalency, there exist precisely two self-orthogonal codes.

Example 8. Let $\mathcal{C}$ be the set of left self-dual codes over $\mathit{E}_3$ of length 4 and type $\left\{ 2, 0 \right\}$ given by the following generator matrix: $\begin{pmatrix} a & 0 & a & a\\ 0 &a & a & e\\ \end{pmatrix}.$

The code in $\mathcal{C}$ has an automorphism group of order 48. Therefore, by Corollary 4,

It demonstrates that, up to monomial equivalency, there exist only one left self-dual code.

Example 9. Let $\mathcal{C}$ be the set of self-dual codes over $\mathit{E}_3$ of length 7 and type $\left\{ 1, 5 \right\}$ given by the following generator matrices:

The codes in $\mathcal{C}$ are inequivalent and they have an automorphism group of order 4608 and 2880 respectively. Therefore, from Corollary 2, we have

which shows that there are exactly two self-dual codes, up to monomial equivalence.

Next, we use our build-up construction methods mentioned in Sections 3.1–3.3 to obtain SO, RSD, LSD, and SD codes over $\mathit{E}_3$. It is possible to discover several self-dual codes of suitable lengths in a fairly effective manner, as demonstrated by our computation. The partial classification results are summarized in Table 22 below.

5.   Conclusions

In this study, we have derived and used propagation rules over a certain non-unital ring of order $9$ to generate self-orthogonal, one-sided self-dual and self-dual codes. Combining this generating technique with mass formulas we have classified these three classes of codes in length at most $7$ up to monomial equivalence. It is an open problem to know if, alike what happens in ^[4], all codes in the three families can be generated by this technique up to some mild type condition.

In order to expand the classification results to longer lengths, more processing power or more efficient automorphism algorithms may be required due to the combinatorial explosion of codes in the three families.

Use of AI tools declaration

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

Acknowledgments

This research work was funded by Institutional Fund Projects under grant No. (IFPRC-105-130-2020). Therefore, authors gratefully acknowledge technical and financial support from the Ministry of Education and King Abdulaziz University, Jeddah, Saudi Arabia.

Conflict of interest

All authors declare no conflicts of interest in this paper.