Self-dual binary $[8m, 4m]$-codes constructed by left ideals of the dihedral group algebra $\mathbb{F}_2[D_{8m}]$

TL;DR

This paper characterizes and counts all self-dual binary codes of length 8m with dihedral symmetry, providing explicit constructions, recursive algorithms, and examples of extremal codes.

Contribution

It offers a complete representation and enumeration of self-dual binary left dihedral codes, including recursive algorithms and a mass formula for counting them.

Findings

Explicit representation of all self-dual binary left dihedral codes.

Recursive algorithms for constructing these codes.

Examples of extremal self-dual codes at lengths 48 and 56.

Abstract

Let $m$ be an arbitrary positive integer and $D_{8m}$ be a dihedral group of order $8m$, i.e., $D_{8m}=\langle x,y\mid x^{4m}=1, y^2=1, yxy=x^{-1}\rangle$. Left ideals of the dihedral group algebra $\mathbb{F}_2[D_{8m}]$ are called binary left dihedral codes of length $8m$, and abbreviated as binary left $D_{8m}$-codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$-codes. These codes make up an important class of self-dual binary $[8m,4m]$-codes such that the dihedral group $D_{8m}$ is necessary a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left $D_{8m}$-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary $[48,24,12]$-code and an extremal self-dual binary $[56,28,12]$-code from self-dual binary left $D_{48}$-codes and left $D_{56}$-codes respectively.

Equations

$$d\leq\left\{\begin{array}[]{ll}4\lfloor\frac{n}{24}\rfloor+6,&{\rm if}\ n\equiv 22\ ({\rm mod}\ 24);\cr 4\lfloor\frac{n}{24}\rfloor+4,&{\rm otherwise}.\end{array}\right.$$

$$\sum _{g\in G}{a}_{g}g+\sum _{g\in G}{b}_{g}g=\sum _{g\in G}(a_g+b_g)g,c(\sum _{g\in G}{a}_{g}g)=\sum _{g\in G}c{a}_{g}g,$$

$$(\sum _{g\in G}{a}_{g}g)(\sum _{g\in G}{b}_{g}g)=\sum _{g\in G}(\sum _{uv=g}{a}_u{b}_v)g,$$

$$D_{2n}$$

$$\Psi (a)=\sum _{i=0}^{n-1}{a}_{i,0}{x}^i+\sum _{i=0}^{n-1}{a}_{i,1}{x}^{i}y.$$

$$4m=2^{\lambda }\cdot m_0\mathrm{and}\lambda ={\lambda }_0+2\ge 2.$$

$$g^{\ast }(x)=(g(x){)}^{\ast }=x^{d}g(x^{-1})=a_d+a_{d-1}x+\dots +a_0{x}^d,$$

$$x^{m_0}-1=\prod _{i=0}^r{f}_i(x),$$

$$w(x)w(x^{-1})\equiv 1(\mathrm{mod}{f}_i(x{)}^s),$$

$$\sum _{j=0}^{2^{\lambda }-1}{a}_j(x)f_i(x{)}^j,\mathrm{where}{a}_j(x)\in {\mathcal{T}}_i,\forall j=0,1,\dots ,2^{\lambda }-1.$$

$$u_i(x)F_i(x)+v_i(x)f_i(x)=1.$$

$$u_i(x^{2^{\lambda }})F_i(x^{2^{\lambda }})+v_i(x^{2^{\lambda }})f_i(x{)}^{2^{\lambda }}=1.$$

$${\epsilon }_i(x)\equiv u_i(x^{2^{\lambda }})F_i(x^{2^{\lambda }})(\mathrm{mod}{x}^{4m}-1).$$

$${\phi }_i:a(x)\mapsto {\epsilon }_i(x)a(x)(\mathrm{mod}{x}^{4m}-1),\forall a(x)\in A_i$$

$$\phi :(a_0(x),\dots ,a_r(x))\mapsto \sum _{i=0}^r{\epsilon }_i(x)a_i(x)(\mathrm{mod}{x}^{4m}-1).$$

$$\{{\epsilon }_i(x),x{\epsilon }_i(x),x^2{\epsilon }_i(x),\dots ,x^{2^{\lambda }{d}_i-1}{\epsilon }_i(x)\}$$

$${\mathrm{w}}_H^{(A_i)}(\xi )=|\{j\mid b_j(x)\ne 0\mathrm{in}{A}_i,j=0,1\}|$$

$$d_H^{(A_i)}(C_i)=\min \{{\mathrm{w}}_H^{(A_i)}(\xi )\mid \xi \ne 0,\xi \in C_i\}.$$

$${\mathcal{A}}_i{\square }_{{\phi }_i}{C}_i$$

$${\dim }_{{\mathbb{F}}_2}({\mathcal{A}}_i{\square }_{{\phi }_i}{C}_i)={\log }_2|C_i|,$$

$$d_H^{({\mathbb{F}}_2)}({\mathcal{A}}_i{\square }_{{\phi }_i}{C}_i)\ge d_H^{({\mathbb{F}}_2)}({\mathcal{A}}_i)\cdot d_H^{(A_i)}(C_i),$$

$$\mathcal{C}=\underset{i=0}{\overset{r}{⨁}}({\mathcal{A}}_i{\square }_{{\phi }_i}{C}_i),$$

$$G_i=(1,a(x)),\mathrm{where}a(x)\in {\mathcal{W}}_i^{(2^{\lambda })}.$$

$$G_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{k}&f_{i}(x)^{k}c(x)\cr 0&f_{i}(x)^{2^{\lambda}-k}\end{array}\right),$$

$${\Omega }_{(\lambda ,d_i)}=1+2^{(2^{\lambda }-1)d_i}+2^{2^{\lambda }{d}_i}+(2^{d_i}+1)\sum _{l=0}^{2^{\lambda }-3}(2^{\lambda }-2-l)2^{l{d}_i}.$$

$$G_i=(1,a(x))\mathrm{and}{G}_{i+ϵ}=(a(x^{-1}),1),$$

$$G_i=(f_i(x)b(x),1)\mathrm{and}{G}_{i+ϵ}=(1,f_{i+ϵ}(x)\cdot x^{-d_i}b(x^{-1})),$$

$$G_i=f_i(x{)}^{2^{\lambda -1}}{I}_2\mathrm{and}{G}_{i+ϵ}=f_{i+ϵ}(x{)}^{2^{\lambda -1}}{I}_2.$$

$$G_{i+\epsilon}=\left(\begin{array}[]{cc}f_{i+\epsilon}(x)^{2^{\lambda}-k-j}c(x^{-1})&f_{i+\epsilon}(x)^{2^{\lambda}-k-j}\cr f_{i+\epsilon}(x)^{2^{\lambda}-k}&0\end{array}\right),$$

$$G_{i+\epsilon}=\left(\begin{array}[]{cc}f_{i+\epsilon}(x)^{2^{\lambda}-k-j}&f_{i+\epsilon}(x)^{2^{\lambda}-k-j}c(x^{-1})\cr 0&f_{i+\epsilon}(x)^{2^{\lambda}-k}\end{array}\right),$$

Full text

I Introduction

The class of self-dual codes is an interesting topic in coding theory due to their connections to other fields of mathematics such as Lattices, Cryptography, Invariant Theory, Block designs, etc. A common theme for the construction of self-dual codes is the use of a computer search. In order to make this search feasible, special construction methods have been used to reduce the search field. In recent years, one of the important construction methods is to use left ideals in a finite group algebra over finite fields and finite rings.

For example, McLoughlin [1] provided a construction of the self-dual, doubly-even and extremal $[48,24,12]$ binary linear block code using a zero divisor in the dihedral group algebra $\mathbb{F}_{2}[D_{48}]$. Dougherty et al. [2] and [3] gave constructions of self-dual and formally self-dual codes from group rings $R[G]$ where the ring $R$ is a finite commutative Frobenius ring. They shown that several of the standard constructions of self-dual codes are found within this general framework. Additionally, they showed precisely which groups can be used to construct the extremal Type II codes of length 24 and 48.

A linear code is said to be self-dual if $C=C^{\bot}$. Binary self-dual codes are called Type II if the weights of all codewords are multiple of $4$ and Type I otherwise. Type II codes are said to have weights that are doubly-even as well. It is well-known that the upper bound for minimum distance $d$ of a binary self-dual code of length $n$ is

[TABLE]

A self-dual binary code is called extremal if it meets the bound.

Let $\mathbb{F}_{q}$ be a finite field of $q$ elements and $G$ be an arbitrary finite group. The group algebra $\mathbb{F}_{q}[G]$ is an $\mathbb{F}_{q}$-algebra with basis $G$. Addition, multiplication with scalars $c\in\mathbb{F}_{q}$ and multiplication are defined by:

[TABLE]

[TABLE]

for any $a_{g},b_{g}\in\mathbb{F}_{q}$ and $g\in G$. Then $\mathbb{F}_{q}[G]$ is a noncommutative ring with identity $1=1_{\mathbb{F}_{q}}1_{G}$ where $1_{\mathbb{F}_{q}}$ and $1_{G}$ is the identity elements of $\mathbb{F}_{q}$ and $G$ respectively. It is known that $\mathbb{F}_{q}[G]$ is semisimple if and only if ${\rm gcd}(q,|G|)=1$.

In this paper, let

[TABLE]

be a dihedral group of order $2n$. For any $a=(a_{0,0},a_{1,0},\ldots$, $a_{n-1,0},a_{0,1},a_{1,1},\ldots,a_{n-1,1})\in\mathbb{F}_{q}^{2n}$, we define

[TABLE]

Then $\Psi$ is an isomorphism of $\mathbb{F}_{q}$-linear spaces from $\mathbb{F}_{q}^{2n}$ onto $\mathbb{F}_{q}[D_{2n}]$. As a natural generalization of Dutra et al. [4], a nonempty subset $\mathcal{C}$ of $\mathbb{F}_{q}^{2n}$ is called a left dihedral code (or left $D_{2n}$-code for more clear) over $\mathbb{F}_{q}$ if $\Psi(\mathcal{C})$ is a left ideal of $\mathbb{F}_{q}[D_{2n}]$. We will equate $\mathcal{C}$ with $\Psi(\mathcal{C})$ in this paper.

There have been many research results on codes as two-sided ideals and left ideals in a finite group algebra over finite fields. For example, Dutra et al [4] investigated codes that are two-sided ideals in a semisimple finite group algebra $\mathbb{F}_{q}[G]$, and given a criterion to decide if these ideals are all the minimal two-sided ideals of $\mathbb{F}_{q}[G]$ when $G$ is a dihedral group. Brochero Martínez [5] showed all central irreducible idempotents and their Wedderburn decomposition of the semisimple dihedral group algebra $\mathbb{F}_{q}[D_{2n}]$ when every divisor of $n$ divides $q-1$. Moreover, we gave a system theory for left $D_{2n}$-codes over finite fields $\mathbb{F}_{q}$ in [6] where ${\rm gcd}(q,n)=1$, and obtained a complete description for left $D_{2n}$-codes over Galois rings ${\rm GR}(p^{2},m)$ in [7] were ${\rm gcd}(p,n)=1$.

One of the most studied open questions in coding theory is to ask whether there is an extremal doubly-even binary self-dual codes of length a multiple of $8$. There are still many problems worth studying in this field. For example,

For which $k$ does there exists a doubly-even self-dual binary $[24k,12k,4k+4]$ code (Open Question 7.7 in [8])?

In this paper, we provide a new way to construct binary self-dual $[8m,4m]$-codes which is different from the methods used in [1], [2], [3] and [4]. Specifically, we give an explicit construction and enumeration for all distinct self-dual binary left $D_{8m}$-codes. In future work, we will try to determine extremal self-dual binary $[8m,4m]$-codes among these codes.

Notation I.1

In this paper, let $\mathbb{F}_{2}=\{0,1\}$ be a binary field and $m=2^{\lambda_{0}}m_{0}$, where $m_{0}$ and $\lambda_{0}$ are nonnegative integers such that $m_{0}$ is odd. Then $8m=2\cdot 4m$ where

[TABLE]

The present paper is organized as follows. In section II, we introduce necessary notations and give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$-codes by Theorem II.4 which is the main result of this paper. In Section III, we give recursive algorithms to solve the problems in the construction of self-dual binary left $D_{8m}$-codes and obtain a clear formula to count the number of all these self-dual codes. In Section IV, we list all distinct self-dual binary left $D_{8m}$-codes for $m=1,3,6,7$. Among these codes, we obtain extremal self-dual binary codes with parameters $[8,4,4]$, $[24,12,8]$, $[48,24,12]$, $[56,28,12]$, respectively. In Section V, we give a detailed proof for Theorem II.4 by four subsections: Give a concatenated structure for every binary left $D_{8m}$-code; Provide a representation and enumeration for all distinct binary left $D_{8m}$-codes; Determine the dual code for each binary left $D_{8m}$-code; Prove Theorem II.4. Section VI concludes the paper.

II Self-dual binary left $D_{8m}$-codes

In this section, we introduce the necessary notations and known results first. Then we give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$-codes.

For any nonzero polynomial $g(x)=\sum_{i=0}^{d}a_{i}x^{i}\in\mathbb{F}_{2}[x]$ of degree $d$, the reciprocal polynomial of $g(x)$ is defined by

[TABLE]

and $g(x)$ is said to be self-reciprocal if $g^{\ast}(x)=g(x)$.

As $m_{0}$ is an odd positive integer, we have that

[TABLE]

where $f_{0}(x),f_{1}(x),\ldots,f_{r}(x)$ are pairwise coprime irreducible polynomials in $\mathbb{F}_{2}[x]$ such that

• •

$r=\rho+2\epsilon$ for some nonnegative integers $\rho$ and $\epsilon$.

• •

$f_{0}(x)=x+1$ with degree $d_{0}=1$.

• •

$f_{i}(x)$ is self-reciprocal and of degree $d_{i}\geq 2$ for all $i=1,\ldots,\rho$.

• •

$f_{\rho+j}(x)$ is not self-reciprocal, $f_{\rho+j}^{\ast}(x)=f_{\rho+j+\epsilon}(x)$ and ${\rm deg}(f_{\rho+j}(x))={\rm deg}(f_{\rho+j+\epsilon}(x))=d_{\rho+j}$ for all $j=1,\ldots,\epsilon$.

It is clear that $m_{0}=\sum_{i=0}^{r}d_{i}$ and $x^{4m}-1=\prod_{i=0}^{r}f_{i}(x)^{2^{\lambda}}.$ This implies $4m=2^{\lambda}\sum_{i=0}^{r}d_{i}$. In this paper, we denote:

$\bullet$ $\mathcal{A}=\mathbb{F}_{2}[x]/\langle x^{4m}-1\rangle$ where we regard elements of $\mathcal{A}$ as polynomials in $\mathbb{F}_{2}[x]$ of degree $<4m$ and the arithmetic is done modulo $x^{4m}-1$.

$\bullet$ $A_{i}=\mathbb{F}_{2}[x]/\langle f_{i}(x)^{2^{\lambda}}\rangle$ where we regard elements of $A_{i}$ as polynomials in $\mathbb{F}_{2}[x]$ of degree $<2^{\lambda}d_{i}$ and the arithmetic is done modulo $f_{i}(x)^{2^{\lambda}}$, for all $i=0,1,\ldots,r$.

Moreover, for any integers $i$ and $s$: $0\leq i\leq\rho$ and $1\leq s\leq 2^{\lambda}$, We adopt the following notation in this paper.

$\bullet$ Let $\mathbb{F}_{2}[x]/\langle f_{i}(x)^{s}\rangle=\{\sum_{j=0}^{sd_{i}-1}a_{j}x^{j}\mid a_{j}\in\mathbb{F}_{2},j=0,1$, $\ldots,sd_{i}-1\}$ in which the arithmetic is done modulo $f_{i}(x)^{s}$.

$\bullet$ Let $\mathcal{W}_{i}^{(s)}$ be the set of elements $w(x)$ in $\mathbb{F}_{2}[x]/\langle f_{i}(x)^{s}\rangle$ satisfying

[TABLE]

where $x^{-1}=x^{4m-1}$ (mod $f_{i}(x)^{s}$).

The rings $A_{i}$ ($0\leq i\leq r$) play important roles in this paper and their structures can be found in many dispersive literature.

Lemma II.1

(cf. [9] Example 2.1) Using the notations above, denote $\langle f_{i}(x)^{s}\rangle=f_{i}(x)^{s}A_{i}$ for any $1\leq s\leq 2^{\lambda}$ and $0\leq i\leq r$. Then we have the following conclusions.

(i) $A_{i}$ is a finite chain ring, $\langle f_{i}(x)\rangle$ is the unique maximal ideal of $A_{i}$, the nilpotency index of $f_{i}(x)$ is $2^{\lambda}$ and $A/\langle f_{i}(x)\rangle\cong\mathbb{F}_{2}[x]/\langle f_{i}(x)\rangle\cong\mathbb{F}_{2^{d_{i}}}$.

(ii) Let ${\cal T}_{i}=\{\sum_{j=0}^{d_{i}-1}t_{j}x^{j}\mid t_{0},t_{1},\ldots,t_{{d_{i}}-1}\in\mathbb{F}_{2}\}\subset A_{i}$. Then every element of $A_{i}$ has a unique $f_{i}(x)$-expansion:

[TABLE]

(iii) We can regard elements of $A_{i}/\langle f_{i}(x)^{s}\rangle$ as the same as the ring $\mathbb{F}_{2}[x]/\langle f_{i}(x)^{s}\rangle$. Hence $|A_{i}/\langle f_{i}(x)^{s}\rangle|=2^{sd_{i}}$.

For each $0\leq i\leq r$, denote $F_{i}(x)=\frac{x^{m_{0}}-1}{f_{i}(x)}\in\mathbb{F}_{2}[x]$. Then $F_{i}(x)$ and $f_{i}(x)$ are coprime polynomials. Hence there are polynomials $u_{i}(x),v_{i}(x)\in\mathbb{F}_{2}[x]$ such that

[TABLE]

This implies $F_{i}(x^{2^{\lambda}})=\frac{x^{4m}-1}{f_{i}(x^{2^{\lambda}})}=\frac{x^{4m}-1}{f_{i}(x)^{2^{\lambda}}}$ and

[TABLE]

In the rest of this paper, let $\varepsilon_{i}(x)\in\mathcal{A}$ satisfying

[TABLE]

From classical ring theory and the Chinese Remainder Theorem, we deduce the following lemma (cf. [9] Lemma 3.2).

Lemma II.2

(i) $\sum_{i=0}^{r}\varepsilon_{i}(x)=1$, $\varepsilon_{i}(x)^{2}=\varepsilon_{i}(x)$ and $\varepsilon_{i}(x)\varepsilon_{j}(x)=0$ for all $0\leq i\neq j\leq r$ in the ring ${\cal A}$.

(ii) ${\cal A}=\bigoplus_{i=0}^{r}{\cal A}_{i}$, where ${\cal A}_{i}=\varepsilon_{i}(x){\cal A}$ with $\varepsilon_{i}(x)$ as its multiplicative identity. Moreover, this decomposition is a ring direct sum in that ${\cal A}_{i}{\cal A}_{j}=\{0\}$ for all $0\leq i\neq j\leq r$.

(iii) For each $0\leq i\leq r$, the map

[TABLE]

is an isomorphism of rings from $A_{i}$ onto ${\cal A}_{i}$.

(iv) For any $a_{i}(x)\in A_{i}$, $0\leq i\leq r$, define

[TABLE]

Then $\varphi$ is a ring isomorphism from the direct product ring $A_{0}\times A_{1}\times\ldots\times A_{r}$ onto $\mathcal{A}$.

As usual, we equate each vector $(a_{0},a_{1},a_{2},\ldots,a_{4m-1})\in\mathbb{F}_{2}^{4m}$ with $a_{0}+a_{1}x+\ldots+a_{4m-1}x^{4m-1}\in\mathcal{A}$. Then binary cyclic codes of length $4m$ are identified with ideals of the ring $\mathcal{A}$. In particular, we have the following properties for the ideal ${\cal A}_{i}=\varepsilon_{i}(x){\cal A}$ of ${\cal A}$.

Corollary II.3

Let $0\leq i\leq r$. Then

(i) ${\cal A}_{i}$ is a binary cyclic code of length $4m$ with parity check polynomial $f_{i}(x)^{2^{\lambda}}$ and generating idempotent $\varepsilon_{i}(x)$.

(ii) As a binary linear code of length $4m$,

[TABLE]

is a basis of ${\cal A}_{i}$. Hence ${\rm dim}_{\mathbb{F}_{2}}({\cal A}_{i})=2^{\lambda}d_{i}$.

Proof:

(ii) Since $A_{i}$ is an $\mathbb{F}_{2}$-linear space with a basis $\{1,x,\ldots,x^{2^{\lambda}d_{i}-1}\}$, by Lemma II.2 (iii) we see that $\{\varepsilon_{i}(x)$, $x\varepsilon_{i}(x),\ldots,x^{2^{\lambda}d_{i}-1}\varepsilon_{i}(x)\}$ is an $\mathbb{F}_{2}$-basis of ${\cal A}_{i}$. ∎

Now, let $C_{i}$ be a linear code of length $2$ over $A_{i}$, i.e. $C_{i}$ is an $A_{i}$-submodule of $A_{i}^{2}=\{(b_{0}(x),b_{1}(x))\mid b_{0}(x),b_{1}(x)\in A_{i}\}$. For each $\xi=(b_{0}(x),b_{1}(x))\in A_{i}^{2}$, we denote by

[TABLE]

the Hamming weight of $\xi$ and define the minimum Hamming distance of $C_{i}$ as

[TABLE]

As a natural generalization of the concept for concatenated codes over finite field (cf. [10], Definition 2.1), using the notations of Lemma II.2 (iii) we define the concatenated code $\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i}$ of the inner code $\mathcal{A}_{i}$ and the outer code $C_{i}$ by

[TABLE]

By Lemma II.2 (iii), we conclude that $\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i}$ is a binary quasi-cyclic code of length $8m$ and index $2$ and the number of codewords is equal to $|\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i}|=|C_{i}|$. This implies

[TABLE]

and the minimum Hamming distance of $\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i}$ satisfies

[TABLE]

where $d_{H}^{(\mathbb{F}_{2})}(\mathcal{A}_{i})$ is the minimum Hamming weight of $\mathcal{A}_{i}$ as a binary linear code of length $4m$.

For the end of this section, we list all distinct self-dual binary left $D_{8m}$-codes by the following theorem.

Theorem II.4

All distinct self-dual binary left $D_{8m}$-codes are given by

[TABLE]

where $C_{i}$ is a linear code of length $2$ over $A_{i}$ with a generator matrix $G_{i}$ given by the following cases:

(${\dagger}$) Let $0\leq i\leq\rho$. Then $G_{i}$ is given by one of the following $1+\sum_{j=1}^{2^{\lambda-1}}|\mathcal{W}_{i}^{(2j)}|$ matrices:

(${\dagger}$-1) $|\mathcal{W}_{i}^{(2^{\lambda})}|$ matrices:

[TABLE]

(${\dagger}$-2) $1$ matrix: $G_{i}=f_{i}(x)^{2^{\lambda-1}}I_{2}$.

(${\dagger}$-3) $\sum_{j=1}^{2^{\lambda-1}-1}|\mathcal{W}_{i}^{(2j)}|$ matrices:

[TABLE]

where $c(x)\in\mathcal{W}_{i}^{(2^{\lambda}-2k)}$ and $1\leq k\leq 2^{\lambda-1}-1$.

(${\ddagger}$) Let $\rho+1\leq i\leq\rho+\epsilon$ and denote

[TABLE]

Then the pair $(G_{i},G_{i+\epsilon})$ is given by one of the following $\Omega_{(\lambda,d_{i})}$ pairs of matrices.

(${\ddagger}$-1) $2^{2^{\lambda}d_{i}}$ pairs:

[TABLE]

where $a(x)\in A_{i}$.

(${\ddagger}$-2) $2^{(2^{\lambda}-1)d_{i}}$ pairs:

[TABLE]

where $b(x)\in\mathbb{F}_{2}[x]/\langle f_{i}(x)^{2^{\lambda}-1}\rangle$.

(${\ddagger}$-3) $1$ pair:

[TABLE]

(${\ddagger}$-4) $\sum_{k=1}^{2^{\lambda}-2}\sum_{j=1}^{2^{\lambda}-k-1}2^{jd_{i}}$ pairs:

G_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{2^{\lambda}-k-j}&f_{i}(x)^{2^{\lambda}-k-j}c(x)\cr 0&f_{i}(x)^{2^{\lambda}-k}\end{array}\right)* and*

[TABLE]

where $c(x)\in\mathbb{F}_{2}[x]/\langle f_{i}(x)^{j}\rangle$, $1\leq j\leq 2^{\lambda}-k-1$ and $1\leq k\leq 2^{\lambda}-2$.

(${\ddagger}$-5) $\sum_{k=1}^{2^{\lambda}-2}\sum_{j=1}^{2^{\lambda}-k-1}2^{(j-1)d_{i}}$ pairs:

G_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{2^{\lambda}-k-j}c(x)&f_{i}(x)^{2^{\lambda}-k-j}\cr f_{i}(x)^{2^{\lambda}-k}&0\end{array}\right)* and*

[TABLE]

where $c(x)\in f_{i}(x)(\mathbb{F}_{2}[x]/\langle f_{i}(x)^{j}\rangle)$, $1\leq j\leq 2^{\lambda}-k-1$ and $1\leq k\leq 2^{\lambda}-2$.

Then the number of self-dual binary left $D_{8m}$-codes is

[TABLE]

To make it convenient for readers, we put a detailed proof for this theorem in Section V.

Now, in order to list self-dual binary left $D_{8m}$-codes by use of Theorem II.4, we have to solve the following problem:

Give an efficient algorithm to determine the set $\mathcal{W}_{i}^{(s)}$ for all integers $s=2j$ and $i$: $1\leq j\leq 2^{\lambda-1}$ and $0\leq i\leq\rho$.

III Recursive algorithm to calculate $\mathcal{W}_{i}^{(s)}$

In this section, we consider how to calculate the sets $\mathcal{W}_{i}^{(s)}$ defined in Section II, for any $1\leq s\leq 2^{\lambda}$ and $0\leq i\leq\rho$.

$\diamondsuit$ $1\leq i\leq\rho$

Let $1\leq i\leq\rho$. Then $d_{i}$ is even since $f_{i}(x)$ is self-reciprocal and irreducible. When $s=1$, we have the following lemma.

Lemma III.1

(cf. [6] Theorem V.5 and its proof) Using the notations above, let $\zeta_{i}(x)$ be a primitive element of the finite field $\mathbb{F}_{2}[x]/\langle f_{i}(x)\rangle$, i.e., the multiplicative order of $\zeta_{i}(x)$ modulo $f_{i}(x)$ is equal to $2^{d_{i}}-1$. Then

(i) $a(x^{-1})\equiv a(x)^{2^{\frac{d_{i}}{2}}}$ $({\rm mod}\ f_{i}(x))$, for any $a(x)\in\mathbb{F}_{2}[x]$.

(ii) The congruence equation $a(x)a(x^{-1})\equiv 1$ $({\rm mod}\ f_{i}(x))$ has exactly $1+2^{\frac{d_{i}}{2}}$ solutions in $\mathbb{F}_{2}[x]/\langle f_{i}(x)\rangle$: $a_{1}(x)=\zeta_{i}(x)^{(2^{\frac{d_{i}}{2}}-1)l}$ for $l=0,1,\ldots,2^{\frac{d_{i}}{2}}$. Hence

[TABLE]

and $|\mathcal{W}_{i}^{(1)}|=2^{\frac{d_{i}}{2}}+1$.

In order to express $\mathcal{W}_{i}^{(s)}$ precisely for $s=2,3,\ldots,2^{\lambda}$, we introduce the following notations:

$\bullet$ $\mathcal{K}_{i}=\mathbb{F}_{2}[x]/\langle f_{i}(x)\rangle$, i.e.,

[TABLE]

in which the arithmetic is done modulo $f_{i}(x)$. Then $\mathcal{K}_{i}$ is a finite field of $2^{d_{i}}$ elements.

$\bullet$ $\mathcal{F}_{i}=\{\xi\in\mathcal{K}_{i}\mid\xi^{2^{\frac{d_{i}}{2}}}=\xi\}$. Then $\mathcal{F}_{i}$ is the unique subfield of $\mathcal{K}_{i}$ with $2^{\frac{d_{i}}{2}}$ elements. Precisely, we have

[TABLE]

$\bullet$ ${\rm Tr}_{\mathcal{K}_{i}/\mathcal{F}_{i}}$ is the trace function from $\mathcal{K}_{i}$ onto $\mathcal{F}_{i}$, i.e.,

[TABLE]

for any $\alpha=a(x)\in\mathcal{K}_{i}$. Then from [11] Corollary 7.17 (i), we deduce the following conclusion.

Lemma III.2

For any $\gamma\in\mathcal{F}_{i}$, we denote

[TABLE]

Then $|{\rm Tr}_{\mathcal{K}_{i}/\mathcal{F}_{i}}^{-1}(\gamma)|=2^{\frac{d_{i}}{2}}$.

We calculate $\mathcal{W}_{i}^{(s)}$ recursively by the following theorem.

Theorem III.3

Let $1\leq i\leq\rho$ and $2\leq s\leq 2^{\lambda}$. Assume $\mathcal{W}_{i}^{(s-1)}$ has been determined. Then $\mathcal{W}_{i}^{(s)}$ can be determined by the following three steps:

Step 1. Choose $a_{s-1}(x)\in\mathcal{W}_{i}^{(s-1)}$ arbitrary, and calculate

[TABLE]

Then $x^{(s-1)\frac{d_{i}}{2}}b_{s-1}(x)\in\mathcal{F}_{i}$.

Step 2. Find $\beta(x)\in\mathcal{K}_{i}=\mathbb{F}_{2}[x]/\langle f_{i}(x)\rangle$ such that $\beta(x)\in{\rm Tr}_{\mathcal{K}_{i}/\mathcal{F}_{i}}^{-1}\left(x^{(s-1)\frac{d_{i}}{2}}b_{s-1}(x)\right)$, i.e.,

[TABLE]

Step 3. Set $a_{1}(x)=a_{s-1}(x)$ $({\rm mod}\ f_{i}(x))$, and calculate

[TABLE]

Then $a_{s}(x)=a_{s-1}(x)+z(x)f_{i}(x)^{s-1}\in\mathcal{W}_{i}^{(s)}$.

Therefore, the number of elements in $\mathcal{W}_{i}^{(s)}$ is equal to

[TABLE]

Proof:

Let $a_{s-1}(x)\in\mathcal{W}_{i}^{(s-1)}$. Then by the definition of $\mathcal{W}_{i}^{(s-1)}$ and $x^{-1}=x^{4m-1}$ in $A_{i}=\mathbb{F}_{2}[x]/\langle f_{i}(x)^{2^{\lambda}}\rangle$, we have

[TABLE]

This implies $f_{i}(x)^{s-1}|\left(a_{s-1}(x)a_{s-1}(x^{4m-1})-1\right)$ in $\mathbb{F}_{2}[x]$, and so $\frac{a_{s-1}(x)a_{s-1}(x^{4m-1})-1}{f_{i}(x)^{s-1}}$ $\in\mathbb{F}_{2}[x]$. Hence the polynomial $b_{s-1}(x)$ calculated by Step 1 belong to $\mathcal{K}_{i}$. As polynomials in $\mathbb{F}_{2}[x]$, by $f_{i}(x^{-1})=x^{-d_{i}}f_{i}^{\ast}(x)=x^{-d_{i}}f_{i}(x)$ we have that

[TABLE]

From this and by Lemma III.1(i), we deduce that

[TABLE]

in $\mathcal{K}_{i}$. This implies $x^{(s-1)\frac{d_{i}}{2}}b_{s-1}(x)\in\mathcal{F}_{i}$, since $\mathcal{F}_{i}$ is a subfield of $\mathcal{K}_{i}$ and $|\mathcal{F}_{i}|=2^{\frac{d_{i}}{2}}$.

Let $a_{s}(x)=a_{s-1}(x)+z(x)f_{i}(x)^{s-1}$ where $z(x)\in\mathcal{K}_{i}$. Then by $a_{s-1}(x)a_{s-1}(x^{4m-1})\equiv 1+b_{s-1}(x)f_{i}(x)^{s-1}$ (mod $f_{i}(x)$), it follows that

[TABLE]

From this we deduce that $a_{s}(x)\in\mathcal{W}_{i}^{(s)}$, i.e. $a_{s}(x)a_{s}(x^{-1})\equiv 1$ (mod $f_{i}(x)^{s}$), if and only if

[TABLE]

(mod $f_{i}(x)$). Then by Lemma III.1 (i), $a_{1}(x)\equiv a_{s-1}(x)$ (mod $f_{i}(x)$) and $a_{1}(x)\in\mathcal{W}_{i}^{(1)}$, we see that the latter condition is equivalent to that $z(x)\in\mathcal{K}_{i}$ satisfying the following condition

[TABLE]

where

[TABLE]

Furthermore, by $a_{1}(x)\in\mathcal{W}_{i}^{(1)}$ and Lemma III.1 we have

[TABLE]

Multiplying $x^{(s-1)\frac{d_{i}}{2}}$ on both sides of Equation (3), we obtain

[TABLE]

Now, set $\beta(x)=\frac{x^{(s-1)\frac{d_{i}}{2}}z(x)}{a_{1}(x)}\in\mathcal{K}_{i}$. Then we have $z(x)=x^{4m-(s-1)\frac{d_{i}}{2}}a_{1}(x)\beta(x)$ (mod $f_{i}(x)$), where $\beta(x)$ satisfies

[TABLE]

i.e., $\beta(x)\in{\rm Tr}_{\mathcal{K}_{i}/\mathcal{F}_{i}}^{-1}\left(x^{(s-1)\frac{d_{i}}{2}}b_{s-1}(x)\right)$.

Finally, by Lemmas III.1 and III.2 it follows that $|\mathcal{W}_{i}^{(s)}|=|\mathcal{W}_{i}^{(s-1)}|\cdot 2^{\frac{d_{i}}{2}}=\ldots=(2^{\frac{d_{i}}{2}}+1)2^{(s-1)\frac{d_{i}}{2}}$. ∎

Therefore, for any $1\leq i\leq\rho$ we have

[TABLE]

From this and by $2^{(2j-1)\frac{d_{i}}{2}}=2^{\frac{d_{i}}{2}}(2^{d_{i}})^{j-1}$, we deduce that

[TABLE]

$\diamondsuit\diamondsuit$ $i=0$

Now, let $i=0$. Then $f_{0}(x)=x+1$,

[TABLE]

and $x^{-1}=x^{2^{\lambda}-1}$ in $A_{0}$. So we have $A_{0}/\langle x+1\rangle=\mathbb{F}_{2}$ and

$\bullet$ $\mathcal{W}_{0}^{(1)}=\{1\}.$

When $s\geq 2$, $\mathcal{W}_{0}^{(s)}$ can be calculated recursively by the following theorem.

Theorem III.4

Let $2\leq s\leq 2^{\lambda}$. For any $a_{s-1}(x)\in\mathcal{W}_{0}^{(s-1)}$, calculate

[TABLE]

Then we have one of the following cases:

(i) If $b_{s-1}=0$, then $a_{s-1}(x),a_{s-1}(x)+(x+1)^{s-1}\in\mathcal{W}_{0}^{(s)}$.

(ii) If $b_{s-1}=1$, then $a_{s-1}(x)\not\in\mathcal{W}_{0}^{(s)}$ and $a_{s-1}(x)+(x+1)^{s-1}\not\in\mathcal{W}_{0}^{(s)}$.

Proof:

By Equation (4), as a polynomial in $\mathbb{F}_{2}[x]$ we have

[TABLE]

Denote $a(x)=a_{s-1}(x)+c_{s-1}(x+1)^{s-1}\in A_{0}/\langle(x+1)^{s}\rangle$ and $u(x)=1+\sum_{t=1}^{2^{\lambda}-1}(x+1)^{t}$, where $c_{s-1}\in\mathbb{F}_{2}$. Then

[TABLE]

This implies $x^{2^{\lambda}-1}=x^{-1}=u(x)$ in $A_{0}$. Therefore, we have

[TABLE]

and $u(x)\equiv u(1)=1$ (mod $x+1$). Hence as a polynomial in $\mathbb{F}_{2}[x]$, we have

[TABLE]

From this and by

[TABLE]

we deduce that

[TABLE]

(i) Let $b_{s-1}=0$. Then

[TABLE]

(mod $(x+1)^{s}$). This implies $a(x)a(x^{-1})=1$ in $A_{0}/\langle(x+1)^{s}\rangle$ by Equation (5). Hence $a(x)=a_{s-1}(x)+c_{s-1}(x+1)^{s-1}\in\mathcal{W}_{0}^{(s)}$ for all $c_{s-1}\in\mathbb{F}_{2}$.

(ii) Let $b_{s-1}=1$. As a polynomial in $\mathbb{F}_{2}[x]$, we have that $a_{s-1}(x)a_{s-1}(x^{2^{\lambda}-1})\equiv 1+(x+1)^{s-1}$ (mod $(x+1)^{s}$). Then by Equation (5), in the ring $A_{0}/\langle(x+1)^{s}\rangle$ we have

[TABLE]

Hence $a(x)=a_{s-1}(x)+c_{s-1}(x+1)^{s-1}\not\in\mathcal{W}_{0}^{(s)}$ for any $c_{s-1}\in\mathbb{F}_{2}$. ∎

Example III.5

Let $\lambda=3$. We calculate $\mathcal{W}_{0}^{(s)}$ for all $s=1,2,3,\ldots,8$. For $a_{1}(x)\in\mathcal{W}_{0}^{(1)}=\{1\}$, we have $a_{1}(x)=1$. As $b_{1}=\frac{a_{1}(x)a_{1}(x^{7})-1}{x+1}=0$, by Theorem III.4 we obtain

$\bullet$* $\mathcal{W}_{0}^{(2)}=\{1,1+(x+1)\}=\{1,x\}$.*

For any $a_{2}(x)\in\mathcal{W}_{0}^{(2)}$, we have $b_{2}=\frac{a_{2}(x)a_{2}(x^{7})-1}{(x+1)^{2}}\equiv 0$ (mod $x+1$). Then by Theorem III.4 it follows that

$\bullet$* $\mathcal{W}_{0}^{(3)}=\{1,1+(x+1)^{2},x,x+(x+1)^{2}\}=\{1,x,x^{2}$, $1+x+x^{2}\}$.*

For any $a_{3}(x)\in\mathcal{W}_{0}^{(3)}$, by a direct calculation we get $b_{3}=\frac{a_{3}(x)a_{3}(x^{7})-1}{(x+1)^{3}}\equiv 0$ (mod $x+1$). By Theorem III.4 we obtain

[TABLE]

For any $a_{4}(x)\in\{1,x,x^{2},x^{3}\}\subset\mathcal{W}_{0}^{(4)}$, it is clear that $b_{4}=\frac{a_{4}(x)a_{4}(x^{7})-1}{(x+1)^{4}}\equiv 0$ (mod $x+1$), but for any $a_{4}(x)\in\{1+x+x^{2},1+x^{2}+x^{3},x+x^{2}+x^{3},1+x+x^{3}\}\subset\mathcal{W}_{0}^{(4)}$ we have $b_{4}=\frac{a_{4}(x)a_{4}(x^{7})-1}{(x+1)^{4}}\equiv 1$ (mod $x+1$). Hence by Theorem III.4 we obtain

[TABLE]

Using a similar method, we get the following calculations:

$\bullet$* $\mathcal{W}_{0}^{(6)}=\{a_{5}(x),a_{5}(x)+(x+1)^{5}\mid a_{5}(x)\in\mathcal{W}_{0}^{(5)}\}$ with $|\mathcal{W}_{0}^{(6)}|=2|\mathcal{W}_{0}^{(5)}|=16=2^{4}.$*

[TABLE]

with $|\mathcal{W}_{0}^{(7)}|=2|\mathcal{W}_{0}^{(5)}|=2^{4}.$

$\bullet$* $\mathcal{W}_{0}^{(8)}=\{a_{7}(x),a_{7}(x)+(x+1)^{8}\mid a_{7}(x)\in\mathcal{W}_{0}^{(7)}\}$ with $|\mathcal{W}_{0}^{(8)}|=2|\mathcal{W}_{0}^{(7)}|=2^{5}.$*

Moreover, we have $|\mathcal{W}_{0}^{(9)}|=2^{5}$ and $|\mathcal{W}_{0}^{(10)}|=|\mathcal{W}_{0}^{(11)}|=2^{6}$ (for $\lambda\geq 4$).

As stated above, we conclude the following conclusion.

Corollary III.6

Using Notation 1.1 in Section I, The number of self-dual binary left $D_{2^{\lambda+1}m_{0}}$-codes is

[TABLE]

where $\omega_{\lambda}=1+\sum_{j=1}^{2^{\lambda-1}}|\mathcal{W}_{0}^{(2j)}|$ and

[TABLE]

In particular, we have the following formulas:

$\diamond$* When $\lambda=2$, The number of self-dual binary left $D_{8m_{0}}$-codes is*

[TABLE]

where $\Omega_{(2,d_{i})}=1+2^{3d_{i}}+2^{4d_{i}}+(2^{d_{i}}+1)(2+2^{d_{i}})$.

$\diamond$* When $\lambda=3$, The number of self-dual binary left $D_{16m_{0}}$-codes is*

[TABLE]

where $\Omega_{(3,d_{i})}=1+2^{7d_{i}}+2^{8d_{i}}+(2^{d_{i}}+1)\sum_{l=0}^{5}(6-l)2^{ld_{i}}$.

Let $SLD(\lambda,m_{0})$ be the number of all self-dual binary left $D_{2^{\lambda+1}m_{0}}$-codes. Then we have the following table:

[TABLE]

Conjecture $|\mathcal{W}_{0}^{(s)}|=2^{1+\lfloor\frac{s}{2}\rfloor}=2\cdot 2^{\lfloor\frac{s}{2}\rfloor}$ for any integer $s\geq 4$. Then

[TABLE]

This conjecture has been proven to hold for $\lambda=2,3$.

IV Self-dual binary left $D_{8m}$-codes for $m=1,3,6,7$

In this section, we describe in detail how to list explicitly all distinct self-dual binary left $D_{8}$-codes, left $D_{24}$-codes, left $D_{48}$-codes and left $D_{56}$-codes, respectively.

Example IV.1

We consider self-dual binary left $D_{8}$-codes. In this case, we have $\lambda=2$, $m=1$ and $x^{4}-1=(x+1)^{4}$ in $\mathbb{F}_{2}[x]$. Then $r=\rho=\epsilon=0$ and $\varepsilon_{0}(x)=1$. By Theorem II.4, all distinct self-dual binary left $D_{8}$-codes are given by: $C_{0}$, where $C_{0}$ is a linear code of length $2$ over the finite chain ring $A_{0}=\mathbb{F}_{2}[x]/\langle(x+1)^{4}\rangle$ with one and only one of the following $11$ matrices as its generator matrix:

(i) $G_{0}=(1,a(x))$, $a(x)\in\mathcal{W}_{0}^{(4)}$;

(ii) $G_{0}=(x+1)^{2}I_{2}$;

(iii) G_{0}=\left(\begin{array}[]{cc}(x+1)&(x+1)c(x)\cr 0&(x+1)^{3}\end{array}\right), $c(x)\in\mathcal{W}_{0}^{(2)}$,

where $\mathcal{W}_{0}^{(4)}$ and $\mathcal{W}_{0}^{(2)}$ are given in Example III.5.

Each self-dual binary left $D_{8}$-code is a self-dual binary $[8,4,d]$-codes where $d$ is the minimal Hamming distance of $C_{0}$. Precisely, we have the following table:

[TABLE]

Example IV.2

We consider binary left $D_{24}$-codes. In this case, $24=2\cdot 12$, $12=4\cdot 3$ and $x^{3}-1=f_{0}(x)f_{1}(x)$ where $f_{0}(x)=x+1$ and $f_{1}(x)=x^{2}+x+1$. Hence $\rho=r=1$, $\epsilon=0$, $d_{0}=1$, $d_{1}=2$ and $x^{12}-1=f_{0}(x)^{4}f_{1}(x)^{4}.$

Let $\mathcal{A}=\mathbb{F}_{2}[x]/\langle x^{12}-1\rangle$, $A_{0}=\mathbb{F}_{2}[x]/\langle(1+x)^{4}\rangle$, $A_{1}=\mathbb{F}_{2}[x]/\langle(1+x+x^{2})^{4}\rangle$, $\varepsilon_{0}(x)=1+x^{4}+x^{8}$ and $\varepsilon_{1}(x)=x^{4}+x^{8}$. By Theorem II.4, all $341$ self-dual binary left $D_{24}$-codes are given by: $\mathcal{C}=(\mathcal{A}_{0}\Box_{\varphi_{0}}C_{0})\oplus(\mathcal{A}_{1}\Box_{\varphi_{1}}C_{1})$, where

$\diamondsuit$* $\mathcal{A}_{0}$ is a binary cyclic code of length $12$ with idempotent generator $\varepsilon_{0}(x)$ and parity check polynomial $1+x^{4}$.*

$\varphi_{0}$* is an isomorphism of rings from $A_{0}$ onto $\mathcal{A}_{0}$ defined by $\varphi_{0}(a(x))=\varepsilon_{0}(x)a(x)$ (mod $x^{12}+1$) for all $a(x)\in A_{0}$.*

$C_{0}$* is one of the $11$ linear codes of length $2$ over $A_{0}$ with a generator matrix $G_{0}$ given in Example IV.1.*

$\diamondsuit$* $\mathcal{A}_{1}$ is a binary cyclic code of length $12$ with idempotent generator $\varepsilon_{1}(x)$ and parity check polynomial $1+x^{4}+x^{8}$.*

$\varphi_{1}$* is an isomorphism of rings from $A_{1}$ onto $\mathcal{A}_{1}$ defined by $\varphi_{1}(b(x))=\varepsilon_{1}(x)b(x)$ (mod $x^{12}+1$) for all $b(x)\in A_{1}$.*

$C_{1}$* is a linear code of length $2$ over $A_{1}$ with a generator matrix $G_{1}$ given by one of the following $31$ matrices:*

(i) $G_{1}=(1,a(x))$, $a(x)\in\mathcal{W}_{1}^{(4)}$;

(ii) $G_{1}=f_{1}(x)^{2}I_{2}$ where $f_{1}(x)=x^{2}+x+1$;

(iii) G_{1}=\left(\begin{array}[]{cc}f_{1}(x)&f_{1}(x)c(x)\cr 0&f_{1}(x)^{3}\end{array}\right), $c(x)\in\mathcal{W}_{1}^{(2)}$,

where $\mathcal{W}_{1}^{(2)}$ and $\mathcal{W}_{1}^{(4)}$ are given in Appendix B.

Among the $341$ codes listed above, we have $24$ self-dual binary $[24,12,8]$-codes with the same weight distribution enumerator $1+759X^{8}+2576X^{12}+759X^{16}+X^{24}$:

[TABLE]

where the generator matrix $G_{0}$ of $C_{0}$ and the generator matrix $G_{1}$ of $C_{1}$ are given by one of the two cases:

(${\dagger}$) $G_{0}=(1,a_{0}(x))$ with $a_{0}(x)\in\{1+x+x^{2},1+x^{2}+x^{3}\}$;

$G_{1}=(1,a_{1}(x))$* with $a_{1}(x)\in\{x+x^{4}+x^{7},x+x^{3}$ $+x^{5}+x^{6},x+x^{2}+x^{6}+x^{7},1+x+x^{3}+x^{5},x^{2}+x^{3}+x^{5}$ $+x^{7},1+x^{3}+x^{4}+x^{5}+x^{7}\}$.*

(${\ddagger}$) $G_{0}=(1,a_{0}(x))$ with $a_{0}(x)\in\{1+x+x^{3},x+x^{2}+x^{3}\}$;

$G_{1}=(1,a_{1}(x))$* with $a_{1}(x)\in\{1+x^{3}+x^{6},x+x^{2}$ $+x^{4}+x^{6},1+x+x^{5}+x^{6},x^{2}+x^{4}+x^{6}+x^{7},1+x^{2}+x^{4}$ $+x^{5},1+x^{2}+x^{3}+x^{4}+x^{7}\}$.*

The above $24$ self-dual binary $[24,12,8]$-codes are extremal and permutation equivalent.

Example IV.3

We consider binary left $D_{48}$-codes. In this case, $48=2\cdot 24$, $24=8\cdot 3$ and $x^{3}-1=f_{0}(x)f_{1}(x)$ where $f_{0}(x)=x+1$ and $f_{1}(x)=x^{2}+x+1$. Hence $r=1$, $d_{0}=1$, $d_{1}=2$ and $x^{24}-1=f_{0}(x)^{8}f_{1}(x)^{8}.$ Let $\mathcal{A}=\mathbb{F}_{2}[x]/\langle x^{24}-1\rangle$, $A_{0}=\mathbb{F}_{2}[x]/\langle(1+x)^{8}\rangle$, $A_{1}=\mathbb{F}_{2}[x]/\langle(1+x+x^{2})^{8}\rangle$,

[TABLE]

By Theorem II.4, all $30149$ self-dual binary left $D_{48}$-codes are given by: $\mathcal{C}=(\mathcal{A}_{0}\Box_{\varphi_{0}}C_{0})\oplus(\mathcal{A}_{1}\Box_{\varphi_{1}}C_{1}),$ here for $i=0,1$:

$\diamond$* $\mathcal{A}_{i}$ is a binary cyclic code of length $24$ with idempotent generator $\varepsilon_{i}(x)$ and parity check polynomial $f_{i}(x)^{8}$.*

$\diamond$* $\varphi_{i}$ is an isomorphism of rings from $A_{i}$ onto $\mathcal{A}_{i}$ defined by $\varphi_{i}(a(x))=\varepsilon_{i}(x)a(x)$ (mod $x^{24}+1$) for all $a(x)\in A_{i}$.*

$\diamond$* $C_{0}$ is a linear code of length $2$ over $A_{0}$ with a generator matrix $G_{0}$ given by one of the following three cases:*

(i) $G_{0}=(1,a(x))$, $a(x)\in\mathcal{W}_{0}^{(8)}$;

(ii) $G_{0}=(x+1)^{4}I_{2}$;

(iii) G_{0}=\left(\begin{array}[]{cc}(x+1)^{k}&(x+1)^{k}c(x)\cr 0&(x+1)^{8-k}\end{array}\right), $c(x)\in\mathcal{W}_{0}^{(8-2k)}$ and $1\leq k\leq 3$,

where $\mathcal{W}_{0}^{(8-2k)}$ is given in Example III.5 for all $k=0,1,2,3$.

$\diamond$* $C_{1}$ is a linear code of length $2$ over $A_{1}$ with a generator matrix $G_{1}$ given by one of the following three cases:*

(i) $G_{1}=(1,b(x))$, $b(x)\in\mathcal{W}_{1}^{(8)}$;

(ii) $G_{1}=f_{1}(x)^{4}I_{2}$ where $f_{1}(x)=x^{2}+x+1$;

(iii) G_{1}=\left(\begin{array}[]{cc}f_{1}(x)^{k}&f_{1}(x)^{k}c(x)\cr 0&f_{1}(x)^{8-k}\end{array}\right), $c(x)\in\mathcal{W}_{1}^{(8-2k)}$ and $1\leq k\leq 3$,

where $\mathcal{W}_{1}^{(2)}$ and $\mathcal{W}_{1}^{(4)}$ are given in Appendix B, $\mathcal{W}_{1}^{(6)}$ and $\mathcal{W}_{1}^{(8)}$ can be calculated easily by use of the algorithm in Theorem III.3. Here we omit the calculation results to save spaces, since $|\mathcal{W}_{1}^{(6)}|=96$ and $|\mathcal{W}_{1}^{(8)}|=384$.

Among the $30149$ codes listed above, we have $192$ doubly-even self-dual binary $[48,24,12]$-codes:

[TABLE]

where $C_{0}$ has the generator matrix $G_{0}=(1,a(x))$, $C_{1}$ has the generator matrix $G_{1}=(1,b(x))$ and the pair $(a(x),b(x))$ is given by Appendix C of this paper. These $192$ self-dual binary $[48,24,12]$-codes are extremal and permutation equivalent.

Example IV.4

We consider binary left $D_{56}$-codes. In this case, $56=2\cdot 28$, $28=4\cdot 7$, $\lambda=2$ and $x^{7}-1=f_{0}(x)f_{1}(x)f_{2}(x)$ where $f_{0}(x)=x+1$, $f_{1}(x)=x^{3}+x+1$ and $f_{2}(x)=x^{3}+x^{2}+1=f^{\ast}_{2}(x)$. Hence $r=2$, $\rho=0$, $\epsilon=1$, $d_{0}=1$, $d_{1}=d_{2}=3$ and $x^{28}-1=f_{0}(x)^{4}f_{1}(x)^{4}f_{2}(x)^{4}.$ Let $\mathcal{A}=\mathbb{F}_{2}[x]/\langle x^{28}-1\rangle$, $A_{0}=\mathbb{F}_{2}[x]/\langle(1+x)^{4}\rangle$, $A_{i}=\mathbb{F}_{2}[x]/\langle f_{i}(x)^{4}\rangle$ for $i=1,2$, $\varepsilon_{0}(x)=x^{24}+x^{20}+x^{16}+x^{12}+x^{8}+x^{4}+1$,

[TABLE]

By Theorem II.4, all $51689$ self-dual binary left $D_{56}$-codes are given by $\mathcal{C}=(\mathcal{A}_{0}\Box_{\varphi_{0}}C_{0})\oplus(\mathcal{A}_{1}\Box_{\varphi_{1}}C_{1})\oplus(\mathcal{A}_{2}\Box_{\varphi_{2}}C_{2}),$ where for $i=0,1,2$ we have the following:

$\diamond$* $\mathcal{A}_{i}$ is a binary cyclic code of length $24$ with idempotent generator $\varepsilon_{i}(x)$ and parity check polynomial $f_{i}(x)^{4}$.*

$\diamond$* $\varphi_{i}$ is an isomorphism of rings from $A_{i}$ onto $\mathcal{A}_{i}$ defined by $\varphi_{i}(a(x))=\varepsilon_{i}(x)a(x)$ (mod $x^{28}+1$) for all $a(x)\in A_{i}$.*

$\diamond$* $C_{0}$ is one of the $11$ linear codes of length $2$ over $A_{0}$ with a generator matrix $G_{0}$ given in Example IV.1.*

$\diamond$* $C_{i}$ is a linear code of length $2$ over $A_{i}$ with a generator matrix $G_{i}$ for $i=1,2$, and the pair $(G_{1},G_{2})$ is given by one of the following $4699$ pairs of matrices where $x^{-1}=x^{27}$ (mod $f_{2}(x)^{4}$):*

(i) $2^{12}=4096$ pairs:

[TABLE]

(ii) $2^{9}=512$ pairs:

[TABLE]

where $b(x)\in A_{1}/\langle f_{1}(x)^{3}\rangle$.

(iii) $1$ pair: $G_{1}=f_{1}(x)^{2}I_{2}\ {\rm and}\ G_{2}=f_{2}(x)^{2}I_{2}.$

(iv) $80$ pairs:

1.* G_{1}=\left(\begin{array}[]{cc}f_{1}(x)^{2}&f_{1}(x)^{2}c(x)\cr 0&f_{1}(x)^{3}\end{array}\right) where $c(x)\in A_{i}/\langle f_{i}(x)\rangle$, and G_{2}=\left(\begin{array}[]{cc}f_{2}(x)^{2}c(x^{-1})&f_{2}(x)^{2}\cr f_{2}(x)^{3}&0\end{array}\right);*

2.* G_{1}=\left(\begin{array}[]{cc}f_{1}(x)&f_{1}(x)c(x)\cr 0&f_{1}(x)^{3}\end{array}\right) where $c(x)\in A_{i}/\langle f_{i}(x)^{2}\rangle$, and G_{2}=\left(\begin{array}[]{cc}f_{2}(x)c(x^{-1})&f_{2}(x)\cr f_{2}(x)^{3}&0\end{array}\right);*

3.* G_{1}=\left(\begin{array}[]{cc}f_{1}(x)&f_{1}(x)c(x)\cr 0&f_{1}(x)^{2}\end{array}\right) where $c(x)\in A_{i}/\langle f_{i}(x)\rangle$, and G_{2}=\left(\begin{array}[]{cc}f_{2}(x)c(x^{-1})&f_{2}(x)\cr f_{2}(x)^{2}&0\end{array}\right).*

(v) $10$ pairs:

1.* G_{1}=\left(\begin{array}[]{cc}0&f_{1}(x)^{2}\cr f_{1}(x)^{3}&0\end{array}\right), G_{2}=\left(\begin{array}[]{cc}f_{2}(x)^{2}&0\cr 0&f_{2}(x)^{3}\end{array}\right);*

2.* G_{1}=\left(\begin{array}[]{cc}f_{1}(x)c(x)&f_{1}(x)\cr f_{1}(x)^{2}&0\end{array}\right) and*

[TABLE]

where $c(x)=f_{1}(x)a(x)$ and $a(x)$ is a polynomial in $\mathbb{F}_{2}[x]$ of degree less than $3$;

3.* G_{1}=\left(\begin{array}[]{cc}0&f_{1}(x)\cr f_{1}(x)^{2}&0\end{array}\right), G_{2}=\left(\begin{array}[]{cc}f_{2}(x)&0\cr 0&f_{2}(x)^{2}\end{array}\right).*

Among $51689$ self-dual binary left $D_{56}$-codes, we have the following $728$ doubly-even self-dual binary $[56,28,12]$-codes:

[TABLE]

where $G_{0}=(1,a(x))$, $G_{1}=(1,\eta(x))$ and $G_{2}=(\eta(x^{-1}),1)$ is a generator matrix of the code $C_{0}$, $C_{1}$ and $C_{2}$, respectively, and the pairs $(a(x),\eta(x))$ of polynomials are given in Appendix D of this paper explicitly. These $728$ self-dual binary $[56,28,12]$-codes are extremal and permutation equivalent.

V Proof of Theorem II.4

In this section, we give a proof for Theorem II.4. It needs to be divided into four parts.

V.1 Concatenated structure of binary left $D_{8m}$-codes

In this subsection, we give a concatenated structure for every binary left $D_{8m}$-code. As

[TABLE]

$C^{(4m)}=\langle x\mid x^{4m}=1\rangle$ is a cyclic subgroup of $D_{8m}$ with order $4m$ generated by $x$. Hence the group algebra $\mathbb{F}_{2}[C^{(4m)}]$ is equal to the residue class ring $\mathcal{A}=\mathbb{F}_{2}[x]/\langle x^{4m}-1\rangle$. This implies that $\mathcal{A}$ is a subring of $\mathbb{F}_{2}[D_{8m}]$ and

[TABLE]

in which $y\alpha(x)=\alpha(x^{-1})y$ for all $\alpha(x)\in\mathcal{A}$. Now, we define a map $\Theta:\mathcal{A}^{2}\rightarrow\mathbb{F}_{2}[D_{8m}]$ by

[TABLE]

Then one can easily verify that $\Theta$ is an $\mathcal{A}$-module isomorphism from $\mathcal{A}^{2}$ onto $\mathbb{F}_{2}[D_{8m}]$.

Let $\mathcal{C}$ be a nonempty subset of $\mathbb{F}_{2}[D_{8m}]$. Then $\mathcal{C}$ is a left ideal of $\mathbb{F}_{2}[D_{8m}]$, i.e. $\mathcal{C}$ is a binary left $D_{8m}$-code, if and only if $\Theta^{-1}(\mathcal{C})$ is an $\mathcal{A}$-submodule of $\mathcal{A}^{2}$ and $y\xi\in\mathcal{C}$ for any $\xi=\alpha(x)+\beta(x)y\in\mathcal{C}$. From this and by

[TABLE]

we deduce that

[TABLE]

Let $\mathcal{C}^{\prime}=\Theta^{-1}(\mathcal{C})$. Then $\mathcal{C}=\Theta(\mathcal{C}^{\prime})$. Hence $\mathcal{C}$ is a binary left $D_{8m}$-code if and only if there is a unique $\mathcal{A}$-submodule $\mathcal{C}^{\prime}$ of $\mathcal{A}^{2}$ satisfying the following condition:

[TABLE]

such that $\Theta(\mathcal{C}^{\prime})=\mathcal{C}$. We will equate $\mathcal{C}$ with $\mathcal{C}^{\prime}$ in this paper.

Theorem V.1

Every binary left $D_{8m}$-code $\mathcal{C}$ can be uniquely decomposed as the following:

[TABLE]

where $C_{i}$, $0\leq i\leq r$, is a linear code of length $2$ over the finite chain ring $A_{i}$ satisfying the following conditions:

(i) If $0\leq i\leq\rho$, $C_{i}$ satisfies

[TABLE]

(ii) If $\rho+1\leq i\leq\rho+\epsilon$, the pair $(C_{i},C_{i+\epsilon})$ of linear codes is given by

[TABLE]

where $C_{i}$ is an arbitrary linear code of length $2$ over $A_{i}$.

Moreover, the number of codewords in $\mathcal{C}$ is $\prod_{i=0}^{r}|C_{i}|$.

Proof:

For any integer $i$, $0\leq i\leq r$, denote

[TABLE]

Now, we claim that

[TABLE]

In fact, by $f_{\mu(i)}(x)=f_{i}^{\ast}(x)=x^{d_{i}}f_{i}(x^{-1})$ we have

[TABLE]

and ${\rm deg}(f_{\mu(i)}(x))={\rm deg}(f_{i}(x))=d_{i}$. From these, we deduce that $f_{i}(x^{-1})^{2^{\lambda}}=x^{-2^{\lambda}d_{i}}f_{\mu(i)}(x)^{2^{\lambda}}$ and

[TABLE]

By Equation (1), it follows that

[TABLE]

where

[TABLE]

[TABLE]

and $\widehat{v}_{\mu(i)}(x),\widehat{u}_{\mu(i)}(x)\in\mathcal{A}$ satisfying:

$\widehat{v}_{\mu(i)}(x)=x^{4m-2^{\lambda}(d_{i}+{\rm deg}(v_{i}(x))}v_{i}^{\ast}(x^{2^{\lambda}})$;

$\widehat{u}_{\mu(i)}(x)=x^{2^{\lambda}d_{i}+4m-2^{\lambda}{\rm deg}(u_{i}(x))}u_{i}^{\ast}(x^{2^{\lambda}})$,

respectively. Hence

[TABLE]

From these and by Equation (2), we deduce that

[TABLE]

Therefore, Equation (7) was proved.

Using the notations in Lemma II.2 (iii) and (iv), for any $(\xi_{i0},\xi_{i1})\in A_{i}^{2}$, $0\leq i\leq r$, we define

[TABLE]

It is clear that $\Phi$ is an $\mathbb{F}_{2}[x]$-module isomorphism from $A_{0}^{2}\times A_{1}^{2}\times\ldots\times A^{2}_{r}$ onto $\mathcal{A}^{2}$. Now, let $\mathcal{C}$ be an $\mathcal{A}$-submodule of $\mathcal{A}^{2}$. Then for each integer $i$, $0\leq i\leq r$, there is a unique $A_{i}$-submodule $C_{i}$ of $A_{i}^{2}$ such that

[TABLE]

It is obvious that $|\mathcal{C}|=\prod_{i=0}^{r}|C_{i}|$.

Moreover, for any integer $i$, $0\leq i\leq r$, and $(a_{i}(x),b_{i}(x))\in C_{i}$, let $(\alpha(x),\beta(x))\in\mathcal{C}$ where

[TABLE]

By Equation (7), in the ring $\mathcal{A}$ we have

$\alpha(x^{-1})=\sum_{i=0}^{r}\varepsilon_{i}(x^{-1})a_{i}(x^{-1})=\sum_{i=0}^{r}\varepsilon_{\mu(i)}(x)a_{i}(x^{-1})$;

$\beta(x^{-1})=\sum_{i=0}^{r}\varepsilon_{i}(x^{-1})b_{i}(x^{-1})=\sum_{i=0}^{r}\varepsilon_{\mu(i)}(x)b_{i}(x^{-1})$.

These imply

[TABLE]

From this and by $\mathcal{C}=\Phi(C_{0}\times C_{1}\times\ldots\times C_{r})$, we deduce that

[TABLE]

Therefore, we have one of the following two cases:

(i) Let $0\leq i\leq\rho$. In this case, we have $\mu(i)=i$ and hence $C_{i}$ satisfies Condition (6).

(ii) Let $\rho+1\leq i\leq\rho+\epsilon$. we have $\mu(i)=i+\epsilon$ and $\mu(i+\epsilon)=i$. In this case, $C_{i}$ and $C_{i+\epsilon}$ satisfy the above conditions if and only if $C_{i+\epsilon}=\{(b_{i}(x^{-1}),a_{i}(x^{-1}))\mid(a_{i}(x),b_{i}(x))\in C_{i}\}$ and $C_{i}$ is an arbitrary linear code over $A_{i}$ of length $2$. ∎

In the rest of this paper, we call ${\cal C}=\bigoplus_{i=1}^{r}(\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i})$ the canonical form decomposition of the binary left $D_{8m}$-code ${\cal C}$.

By Theorem V.1, in order to list all distinct binary left $D_{8m}$-codes we just need to solve the following questions:

Question 1. Determine all linear codes of length $2$ over $A_{i}$ for each $i=\rho+1,\ldots,\rho+\epsilon$.

Question 2. Determine all linear codes of length $2$ over $A_{i}$ satisfying Condition (6) for each $i=0,1,\ldots,\rho$.

V.2 Representation and enumeration for binary left $D_{8m}$-codes

In this subsection, we solve the two questions at the end of Subsection V.1 first. Then we obtain an explicit representation and enumeration for all distinct binary left $D_{8m}$-codes.

Let $0\leq i\leq r$ and $a(x)\in A_{i}=\mathbb{F}_{2}[x]/\langle f_{i}(x)^{2^{\lambda}}\rangle$. By Lemma II.1(ii), $a(x)$ has a unique $f_{i}(x)$-adic expansion:

[TABLE]

If $a(x)\neq 0$, we define the $f_{i}(x)$-degree of $a(x)$ as the least index $j$ for which $a_{j}(x)\neq 0$ and denote as $\|a(x)\|_{f_{i}(x)}=j$. If $\alpha(x)=0$ we define $\|a(x)\|_{f_{i}(x)}=2^{\lambda}$.

Furthermore, for any vector $\alpha=(a(x),b(x))\in A_{i}^{2}$, where $a(x),b(x)\in A_{i}$, we define the $f_{i}(x)$-degree of $\alpha$ by

[TABLE]

Let $1\leq j\leq 2^{\lambda}$ and $b(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle=\mathbb{F}_{2}[x]/\langle f_{i}(x)^{j}\rangle$. By Lemma II.1 (iii), $b(x)$ has a unique $f_{i}(x)$-expansion:

[TABLE]

where $b_{l}(x)\in\mathcal{T}_{i}$ for all $l=0,1,\ldots,j-1$. So

[TABLE]

(cf. [12] or [13]). In the rest of this paper, we will identify the element $b(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$ with $\sum_{l=0}^{j-1}f_{i}(x)^{l}b_{l}(x)\in A_{i}$ and regard $A_{i}/\langle f_{i}(x)^{j}\rangle$ as a subset of $A_{i}$.

Let $C_{i}$ be a linear code over $A_{i}$ of length $2$. By [12] Definition 3.1, a matrix $G_{i}$ is called a generator matrix for $C_{i}$ if the rows of $G_{i}$ span $C_{i}$ and none of them can be written as an $A_{i}$-linear combination of the other rows of $G_{i}$.

$\diamondsuit$ First, for Question 1 at the end of Subsection V.1 we have the following conclusion:

Lemma V.2

(cf. [9] Example 2.5) Using the notation above, let $0\leq i\leq r$. Then the number of linear codes over $A_{i}$ of length $2$ is equal to

[TABLE]

Precisely, every linear code $C_{i}$ over $A_{i}$ of length $2$ has one and only one of the following matrices $G_{i}$ as its generator matrices in standard form:

1.* $G_{i}=(1,a(x))$, where $a(x)\in A_{i}$.*

2.* $G_{i}=(f_{i}(x)^{k},f_{i}(x)^{k}a(x))$, where $1\leq k\leq 2^{\lambda}-1$ and $a(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-k}\rangle$.*

3.* $G_{i}=(f_{i}(x)b(x),1)$, where $b(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-1}\rangle$.*

4.* $G_{i}=(f_{i}(x)^{k+1}b(x),f_{i}(x)^{k})$, where $1\leq k\leq 2^{\lambda}-1$ and $b(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-k-1}\rangle$.*

5.* G_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{k}&0\cr 0&f_{i}(x)^{k}\end{array}\right), where $0\leq k\leq 2^{\lambda}$.*

6.* G_{i}=\left(\begin{array}[]{cc}1&c(x)\cr 0&f_{i}(x)^{j}\end{array}\right), where $1\leq j\leq 2^{\lambda}-1$ and $c(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$.*

7.* G_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{k}&f_{i}(x)^{k}c(x)\cr 0&f_{i}(x)^{k+j}\end{array}\right), where $1\leq j\leq 2^{\lambda}-k-1$, $1\leq k\leq 2^{\lambda}-2$ and $c(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$.*

8.* G_{i}=\left(\begin{array}[]{cc}c(x)&1\cr f_{i}(x)^{j}&0\end{array}\right), where $1\leq j\leq 2^{\lambda}-1$ and $c(x)\in f_{i}(x)(A_{i}/\langle f_{i}(x)^{j}\rangle)$.*

9.* G_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{k}c(x)&f_{i}(x)^{k}\cr f_{i}(x)^{k+j}&0\end{array}\right), where $1\leq j\leq 2^{\lambda}-k-1$, $1\leq k\leq 2^{\lambda}-2$ and $c(x)\in f_{i}(x)(A_{i}/\langle f_{i}(x)^{j}\rangle)$.*

Then from [12] Proposition 3.2 and Theorem 3.5, we deduce the following.

Lemma V.3

Let $C_{i}$ be a linear code over $A_{i}$ of length $2$ with generator matrix $G_{i}$.

(i) If $G_{i}$ is given by Cases 1–4 in Lemma V.2 and let $\|G_{i}\|_{f_{i}(x)}=t$, then the number of codewords in $C_{i}$ is equal to $|C_{i}|=|\mathcal{T}_{i}|^{2^{\lambda}-t}=2^{(2^{\lambda}-t)d_{i}}$. In this case, we have

[TABLE]

(ii) Let $G_{i}$ be given by Cases 5–9 in Lemma V.2 and assume that $\alpha_{1},\alpha_{2}$ are the two row vectors of $G_{i}$. If $\|\alpha_{k}\|_{f_{i}(x)}=t_{k}$ for $k=1,2$, the number of codewords in $C_{i}$ is equal to $|C_{i}|=|\mathcal{T}_{i}|^{(2^{\lambda}-t_{1})+(2^{\lambda}-t_{2}))}=2^{(2^{\lambda+1}-t_{1}-t_{2})d_{i}}$. In this case, we have

[TABLE]

$\diamondsuit\diamondsuit$ Then we solve Question 2 at the end of Subsection V.1. To do this we need the following lemma.

Lemma V.4

Let $0\leq i\leq r$. Then

(i) $x$ is an invertible element of $A_{i}$ satisfying $x^{-1}=x^{4m-1}$ $({\rm mod}\ f_{i}(x)^{2^{\lambda}})$.

(ii) In the ring $A_{\mu(i)}$, we have

[TABLE]

where $x^{-kd_{i}}=x^{4m-kd_{i}}$ $({\rm mod}\ f_{\mu(i)}(x)^{2^{\lambda}})$, for any $1\leq k\leq 2^{\lambda}-1$. Specifically, we have

[TABLE]

Proof:

(i) Since $f_{i}(x)^{2^{\lambda}}$ is a divisor of $x^{4m}-1$ in $\mathbb{F}_{2}[x]$, we have that $x^{4m}=1$, i.e. $x^{-1}=x^{4m-1}$ in $A_{i}$.

(ii) As $f_{i}(x)$ is an irreducible divisor of $x^{m_{0}}-1$ with degree $d_{i}$, by (i) we see that $f_{i}(x^{-1})^{k}=x^{4m-kd_{i}}(x^{d_{i}}f_{i}(x^{-1}))^{k}=x^{4m-kd_{i}}f_{i}^{\ast}(x)^{k}=x^{4m-kd_{i}}f_{\mu(i)}(x)^{k}$ in $A_{\mu(i)}$. ∎

For $A_{i}$-linear codes $C_{i}$ of length $2$ satisfying Condition (6) in Theorem V.1, we have the following theorem.

Theorem V.5

Let $0\leq i\leq\rho$. Then all distinct linear codes $C_{i}$ of length $2$ over $A_{i}$ satisfying Condition (6) in Theorem V.1 (i) are given by the following five cases, where $G_{i}$ is a generator matrix of the code $C_{i}$.

I. $|\mathcal{W}_{i}^{(2^{\lambda})}|$ codes:

[TABLE]

II. $\sum_{k=1}^{2^{\lambda}-1}|\mathcal{W}_{i}^{(2^{\lambda}-k)}|$ codes:

[TABLE]

where $a(x)\in\mathcal{W}_{i}^{(2^{\lambda}-k)}$ and $1\leq k\leq 2^{\lambda}-1$

III. $2^{\lambda}+1$ codes:

[TABLE]

where $0\leq k\leq 2^{\lambda}$ and $I_{2}$ is the identity matrix of order $2$.

IV. $\sum_{j=1}^{2^{\lambda}-1}|\mathcal{W}_{i}^{(j)}|$ codes:

[TABLE]

where $c(x)\in\mathcal{W}_{i}^{(j)}$ and $1\leq j\leq 2^{\lambda}-1$.

V. $\sum_{k=1}^{2^{\lambda}-2}\sum_{j=1}^{2^{\lambda}-k-1}|\mathcal{W}_{i}^{(j)}|$ codes:

[TABLE]

where $c(x)\in\mathcal{W}_{i}^{(j)}$, $1\leq j\leq 2^{\lambda}-k-1$ and $1\leq k\leq 2^{\lambda}-2$.

Therefore, the number of linear codes of length $2$ over $A_{i}$ satisfying Condition (6) in Theorem V.1 (i) is equal to

[TABLE]

Proof:

See Appendix A. ∎

Finally, from Theorems V.1 and V.5 we deduce the following corollary.

Corollary V.6

Every binary left $D_{8m}$-code $\mathcal{C}$ can be constructed by the following three steps:

(i) For each $i=0,1,\ldots,\rho$, choose a linear code $C_{i}$ of length $2$ over $A_{i}$ listed in Theorem V.5

(ii) For each $i=\rho+1,\ldots,\rho+\epsilon$, choose a linear code $C_{i}$ of length $2$ over $A_{i}$ listed in Lemma V.2 and set

[TABLE]

(iii) Set

[TABLE]

Moreover, the number of codewords in the binary left $D_{8m}$-code $\mathcal{C}$ constructed above is equal to $|{\cal C}|=\prod_{i=0}^{r}|C_{i}|,$ where $|C_{i}|$ is determined by Lemma V.3 for all $i=0,1,\ldots,r$.

Hence the number of all binary left $D_{8m}$-codes is

[TABLE]

Therefore, in order to list all distinct binary left $D_{8m}$-codes explicitly, we need to determine the subset $\mathcal{W}_{i}^{(s)}$ of $\mathbb{F}_{2}[x]/\langle f_{i}(x)^{s}\rangle$ precisely, for all $1\leq s\leq 2^{\lambda}$ and $0\leq i\leq\rho$. This work has done at Section III.

V.3 The dual code of every binary left $D_{8m}$-code

In this subsection, we determine the dual code of each binary left $D_{8m}$-code.

Let $\textit{{a}}=(a_{0,0},a_{1,0},\ldots,a_{4m-1,0},a_{0,1},a_{1,1},\ldots,a_{4m-1,1})$, $\textit{{b}}=(b_{0,0},b_{1,0},\ldots,b_{4m-1,0},b_{0,1},b_{1,1},\ldots,b_{4m-1,1})\in\mathbb{F}_{2}^{8m}$. The inner product of a and b is defined by

[TABLE]

Let $C$ be a binary linear code of length $8m$, i.e. a subspace of $\mathbb{F}_{2}^{8m}$. Then the dual code of $C$ is defined by

[TABLE]

and $C$ is said to be self-dual if $C=C^{\bot}$.

For any $a(x)=\sum_{i=0}^{4m-1}a_{i}x^{i}\in\mathcal{A}$, by $x^{4m}=1$ we have $a(x^{-1})=a_{0}+\sum_{i=1}^{4m-1}a_{i}x^{4m-i}\in\mathcal{A}$. Let $G=\left(g_{ij}(x)\right)_{k\times l}$ be a matrix over $A_{i}$ of size $k\times l$ with $g_{ij}(x)\in A_{i}$. We define

[TABLE]

and denote by $G^{{\rm tr}}$ the transpose of $G$, i.e., $G^{{\rm tr}}=\left(h_{ij}(x)\right)_{l\times k}$ where $h_{ij}(x)=g_{ji}(x)$.

Lemma V.7

Using the notation above, denote

[TABLE]

Then $[\textbf{a},\textbf{b}]=0$ if $(a_{0}(x),a_{1}(x))\cdot(b_{0}(x^{-1}),b_{1}(x^{-1}))^{\rm tr}=0$ in the ring $\mathcal{A}=\mathbb{F}_{2}[x]/\langle x^{4m}-1\rangle$.

Proof:

By $x^{4m}=1$ in $\mathcal{A}$, there exist $h_{1},\ldots,h_{4m-1}\in\mathbb{F}_{2}$ such that

[TABLE]

Therefore, $(a_{0}(x),a_{1}(x))\cdot(b_{0}(x^{-1}),b_{1}(x^{-1}))^{\rm tr}=0$ in $\mathcal{A}$ implies that $[\textbf{a},\textbf{b}]=0$. ∎

Now, we give the dual code of each binary left $D_{8m}$-code.

Theorem V.8

Let $\mathcal{C}$ be a binary left $D_{8m}$-code with canonical form decomposition ${\cal C}=\bigoplus_{i=1}^{r}(\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i})$, where $C_{i}$ is a linear code of length $2$ over $A_{i}$ with a generator matrix $G_{i}$. Then the dual code of ${\cal C}$ is a binary left $D_{8m}$-code with the following canonical form decomposition:

[TABLE]

where $Q_{i}$ is a linear code of length $2$ over $A_{i}$ with a generator matrix $H_{i}$ given by the following two cases:

(${\dagger}$) Let $0\leq i\leq\rho$. Then $H_{i}$ is given by one the following five subcases:

(${\dagger}$-I) $H_{i}=(1,a(x))$, if $G_{i}$ is given by Case I in Theorem V.5.

(${\dagger}$-II) H_{i}=\left(\begin{array}[]{cc}1&a(x)\cr 0&f_{i}(x)^{2^{\lambda}-\kappa}\end{array}\right), if $G_{i}$ is given by Case II in Theorem V.5.

(${\dagger}$-III) $H_{i}=f_{i}(x)^{2^{\lambda}-\kappa}I_{2}$, if $G_{i}$ is given by Case III in Theorem V.5.

(${\dagger}$-IV) $H_{i}=(f_{i}(x)^{2^{\lambda}-j},f_{i}(x)^{2^{\lambda}-j}c(x))$, if $G_{i}$ is given by Case IV in Theorem V.5.

(${\dagger}$-V) H_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{2^{\lambda}-k-j}&f_{i}(x)^{2^{\lambda}-k-j}c(x)\cr 0&f_{i}(x)^{2^{\lambda}-k}\end{array}\right), if $G_{i}$ is given by Case V in Theorem V.5. (${\ddagger}$) Let $\rho+1\leq i\leq\rho+\epsilon$. Then the pair $(H_{i},H_{i+\epsilon})$ of matrices is given by one the following nine subcases:

(${\ddagger}$-1) If $G_{i}$ is given by Case 1 in Lemma V.2,

[TABLE]

(${\ddagger}$-2) If $G_{i}$ is given by Case 2 in Lemma V.2,

[TABLE]

where $a(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-k}\rangle$ and $1\leq k\leq 2^{\lambda}-1$.

(${\ddagger}$-3) If $G_{i}$ is given by Case 3 in Lemma V.2,

[TABLE]

where $b(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-1}\rangle$.

(${\ddagger}$-4) if $G_{i}$ is given by Case 4 in Lemma V.2,

[TABLE]

where $b(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-k-1}\rangle$ and $1\leq k\leq 2^{\lambda}-1$.

(${\ddagger}$-5) If $G_{i}$ is given by Case 5 in Lemma V.2,

H_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{2^{\lambda}-k}&0\cr 0&f_{i}(x)^{2^{\lambda}-k}\end{array}\right)* and*

H_{i+\epsilon}=\left(\begin{array}[]{cc}f_{i+\epsilon}(x)^{2^{\lambda}-k}&0\cr 0&f_{i+\epsilon}(x)^{2^{\lambda}-k}\end{array}\right),**

where $0\leq k\leq 2^{\lambda}$.

(${\ddagger}$-6) If $G_{i}$ is given by Case 6 in Lemma V.2,

$H_{i}=(f_{i}(x)^{2^{\lambda}-j},f_{i}(x)^{2^{\lambda}-j}c(x))$* and*

$H_{i+\epsilon}=(f_{i+\epsilon}(x)^{2^{\lambda}-j}c(x^{-1}),f_{i+\epsilon}(x)^{2^{\lambda}-j}),$**

where $c(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$ and $1\leq j\leq 2^{\lambda}-1$.

(${\ddagger}$-7) If $G_{i}$ is given by Case 7 in Lemma V.2,

H_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{2^{\lambda}-k-j}&f_{i}(x)^{2^{\lambda}-k-j}c(x)\cr 0&f_{i}(x)^{2^{\lambda}-k}\end{array}\right)* and*

[TABLE]

where $c(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$, $1\leq j\leq 2^{\lambda}-k-1$ and $1\leq k\leq 2^{\lambda}-2$.

(${\ddagger}$-8) If $G_{i}$ is given by Case 8 in Lemma V.2,

$H_{i}=(f_{i}(x)^{2^{\lambda}-j+1}c(x),f_{i}(x)^{2^{\lambda}-j})$* and*

$H_{i+\epsilon}=(f_{i+\epsilon}(x)^{2^{\lambda}-j},f_{i+\epsilon}(x)^{2^{\lambda}-j+1}\cdot x^{-d_{i}}c(x^{-1})),$**

where $c(x)\in f_{i}(x)(A_{i}/\langle f_{i}(x)^{j}\rangle)$ and $1\leq j\leq 2^{\lambda}-1$.

(${\ddagger}$-9) If $G_{i}$ is given by Case 9 in Lemma V.2,

H_{i}=\left(\begin{array}[]{cc}f_{i}(x)^{2^{\lambda}-k-j}c(x)&f_{i}(x)^{2^{\lambda}-k-j}\cr f_{i}(x)^{2^{\lambda}-k}&0\end{array}\right)* and*

[TABLE]

where $c(x)\in f_{i}(x)(A_{i}/\langle f_{i}(x)^{j}\rangle)$, $1\leq j\leq 2^{\lambda}-k-1$ and $1\leq k\leq 2^{\lambda}-2$.

Proof:

For any integer $i$, $0\leq i\leq r$, let $Q_{i}$ be the linear code of length $2$ over $A_{i}$ with $H_{i}$ as its generator matrix. When $0\leq i\leq\rho$, by Theorem V.5 we see that $Q_{i}$ is an $A_{i}$-submodule of $A_{i}^{2}$ satisfying Condition (6) and $|C_{i}||Q_{i}|=2^{2\cdot 2^{\lambda}d_{i}}$. When $\rho+1\leq i\leq\rho+2\epsilon$, by Lemmas V.2 and V.3 we see that $Q_{i}$ is an $A_{i}$-submodule of $A_{i}^{2}$ satisfying $|C_{i}||Q_{i}|=2^{2\cdot 2^{\lambda}d_{i}}$ and

[TABLE]

for all $l=\rho+1,\ldots,\rho+\epsilon$. Now, we set $\mathcal{Q}=\bigoplus_{i=0}^{r}(\mathcal{A}_{i}\Box_{\varphi_{i}}Q_{i}).$ Then by Theorem V.1, we conclude that $\mathcal{Q}$ is a binary left $D_{8m}$-code. Moreover, by $4m=2^{\lambda}\sum_{i=0}^{r}d_{i}$ and $|{\cal C}||\mathcal{Q}|=(\prod_{i=0}^{r}|C_{i}|)(\prod_{i=0}^{r}|Q_{i}|)=\prod_{i=0}^{r}|C_{i}||Q_{i}|$, we have

[TABLE]

Let $(a_{0}(x),a_{1}(x))\in\mathcal{C}$ and $(b_{0}(x),b_{1}(x))\in\mathcal{Q}$. Then there exist $\xi_{i}\in A_{i}$ or $\xi_{i}\in A^{2}$ and $\eta_{i}\in A_{i}$ or $\eta_{i}\in A_{i}^{2}$ such that

$(a_{0}(x),a_{1}(x))=\sum_{i=0}^{r}\varepsilon_{i}(x)\xi_{i}G_{i}\in\mathcal{A}^{2},$

$(b_{0}(x),b_{1}(x))=\sum_{i=0}^{r}\varepsilon_{i}(x)\eta_{i}H_{i}\in\mathcal{A}^{2}$.

Then from Lemma II.2(i), Equation (7) and

[TABLE]

we deduce that

[TABLE]

Now, we have the following two cases.

Case (${\dagger}$) Let $0\leq i\leq\rho$. By Theorems V.5, we have one of the following subcases:

Subcase (${\dagger}$-I) Let $G_{i}=(1,a(x))$ and $H_{i}=(1,a(x))$, where $a(x)\in\mathcal{W}_{i}^{(2^{\lambda})}$. By $a(x)\in\mathcal{W}_{i}^{(2^{\lambda})}$, it follows that $a(x)\in A_{i}$ satisfying $a(x)a(x^{-1})=1$. Hence $\mu(H_{i})=(1,a(x^{-1}))$ and so $G_{i}\cdot(\mu(H_{i}))^{{\rm tr}}=1+a(x)a(x^{-1})=0.$

Subcase (${\dagger}$-II) Let $G_{i}=(f_{i}(x)^{k},f_{i}(x)^{k}a(x))$ and H_{i}=\left(\begin{array}[]{cc}1&a(x)\cr 0&f_{i}(x)^{2^{\lambda}-k}\end{array}\right), where $a(x)\in\mathcal{W}_{i}^{(2^{\lambda}-k)}$ and $1\leq k\leq 2^{\lambda}-1$. Hence (\mu(H_{i}))^{{\rm tr}}=\left(\begin{array}[]{cc}1&0\cr a(x^{-1})&f_{i}(x^{-1})^{2^{\lambda}-k}\end{array}\right). By $a(x)\in\mathcal{W}_{i}^{(2^{\lambda}-k)}$, it follows that $f_{i}(x)^{k}=f_{i}(x)^{k}a(x)a(x^{-1})$. By Lemma V.4(ii), we have

[TABLE]

Then from a direct calculation, we obtain $G_{i}\cdot(\mu(H_{i}))^{{\rm tr}}=0$.

One can easily verify that $G_{i}\cdot(\mu(H_{i}))^{{\rm tr}}=0$ for Subcases (${\dagger}$-III), (${\dagger}$-IV) and (${\dagger}$-V). Here, we omit the proofs.

Case (${\ddagger}$) Let $\rho\leq i\leq\rho+2\epsilon$. Then by Theorem V.1, one can easily verify that $G_{i}\cdot(\mu(H_{i+\epsilon}))^{{\rm tr}}=0$ for all $i=\rho+1,\ldots,\rho+\epsilon$, and $G_{i}\cdot(\mu(H_{i-\epsilon}))^{{\rm tr}}=0$ for all $i=\rho+\epsilon+1,\ldots,\rho+2\epsilon$.

Therefore, $(a_{0}(x),a_{1}(x))\cdot(b_{0}(x^{-1}),b_{1}(x^{-1}))^{{\rm tr}}=0$ for any $(a_{0}(x),a_{1}(x))\in\mathcal{C}$ and $(b_{0}(x),b_{1}(x))\in\mathcal{Q}$. Hence $\mathcal{Q}\subseteq\mathcal{C}^{\bot}$ by Lemma V.7. From this and by Equation (8), we deduce that $\mathcal{C}^{\bot}=\mathcal{Q}$ as required. ∎

V.4 Proving Theorem II.4

We give a proof of Theorem II.4 as follows:

Using the notations of Theorems V.1, V.5 and V.8, let

[TABLE]

where $C_{i}$ and $Q_{i}$ are $A_{i}$-submodules of $A_{i}^{2}$ determined by Theorem V.8. From this and by Theorem V.1, we deduce that ${\cal C}={\cal C}^{\bot}$ if and only if for any integer $i$, $0\leq i\leq\rho+\epsilon$, we have that $C_{i}=Q_{i}$. The latter is equivalent to $G_{i}=H_{i}$ by Lemma V.2. Hence by $\lambda\geq 2$ we deduce the following conclusions:

(${\dagger}$) Let $0\leq i\leq\rho$. We only need to consider subcase (${\dagger}$-V) in Theorem V.8. In this case, $G_{i}=H_{i}$ if and only if $k=2^{\lambda}-k-j$, i.e. $j=2^{\lambda}-2k$. This implies $k+j=2^{\lambda}-k$. Moreover, by $j\geq 1$ we conclude that $1\leq k\leq 2^{\lambda-1}-1$.

(${\ddagger}$) Let $\rho+1\leq i\leq\rho+\epsilon$. Then the conclusions follows from $G_{i}=H_{i}$ and Theorem V.8(${\ddagger}$) immediately. Moreover, the number of pairs $(G_{i},G_{i+\epsilon})$ is

[TABLE]

which is equal to $\Omega_{(\lambda,d_{i})}$.

Remark For each self-dual binary left $D_{8m}$-code $\mathcal{C}=\bigoplus_{i=0}^{r}(\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i})$ listed by Theorem II.4, a generator matrix of $\mathcal{C}$ is given by G_{\mathcal{C}}=\left(\begin{array}[]{c}B_{0}\cr B_{1}\cr\ldots\cr B_{r}\end{array}\right), where $B_{i}$ is a generator mathix of the subcode $\mathcal{A}_{i}\Box_{\varphi_{i}}C_{i}$, and $B_{i}$ can be easily determined by Lemma V.3 and the definition of concatenated codes in Section II for all $i=0,1,\ldots,r$.

VI Conclusion and further work

Self-dual binary left $D_{8m}$-codes make up an important class of self-dual binary $[8m,4m]$-codes such that the dihedral group $D_{8m}$ is necessary a subgroup of the automorphism group of each code. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$-codes. In particular, we provide recursive algorithms to solve problems in the construction of these codes and obtain a precise formula to count the number of all these codes. In order to enable readers to use the results of the paper directly to construct self-dual binary $[8m,4m]$-codes, we give a detailed descriptions of the generator matrices for each self-dual binary left $D_{8m}$-code.

Future topics of interest include to determine extremal binary self-dual codes of length $8m$ among self-dual binary left $D_{8m}$-codes, and consider the existence of self-dual, doubly-even and extremal binary linear codes with basic parameters $[24k,12k,4k+4]$ which are also self-dual binary left $D_{24k}$-codes for some integers $k\geq 3$.

Acknowledgment

Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 11801324, 61571243, 11701336), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007), the Scientific Research Foundation for the PhD of Shandong University of Technology (Grant No. 417037), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University) (Grant No. AM201804) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09).

Appendix A: Proof of Theorem V.5

Let $C_{i}$ be a linear codes $C_{i}$ of length $2$ over $A_{i}$, where $0\leq i\leq\rho$. By Lemma V.2, $C_{i}$ has one and only one of the following matrices $G$ as its generator matrix:

Cases 1–2 $G=(f_{i}(x)^{k},f_{i}(x)^{k}a(x))$, where $0\leq k\leq 2^{\lambda}-1$ and $a(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-k}\rangle$. In this case, $C_{i}$ satisfies Condition (6) if and only if there exists $u(x)\in A_{i}$ such that

[TABLE]

which is equivalent to that $f_{i}(x^{-1})^{k}a(x^{-1})=u(x)f_{i}(x)^{k}$ and $f_{i}(x^{-1})^{k}=u(x)f_{i}(x)^{k}a(x)$. Then we have $f_{i}(x^{-1})^{k}=f_{i}(x^{-1})^{k}a(x^{-1})a(x)$, where $f_{i}(x^{-1})^{k}=x^{-kd_{i}}f_{i}(x)^{k}$ by Lemma V.4(ii). From these we deduce that $C_{i}$ satisfies Condition (6) if and only if $a(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-k}\rangle$ satisfying

[TABLE]

The latter is equivalent to $a(x)a(x^{-1})\equiv 1\ ({\rm mod}\ f_{i}(x)^{2^{\lambda}-k})$, i.e., $a(x)\in\mathcal{W}_{i}^{(2^{\lambda}-k)}$.

By $\|(f_{i}(x)^{k},f_{i}(x)^{k}a(x))\|_{f_{i}(x)}=k$ and Lemma V.3(i), it follows that $|C_{i}|=2^{(2^{\lambda}-k)d_{i}}$.

Cases 3–4 $G=(f_{i}(x)^{k+1}b(x),f_{i}(x)^{k})$, where $0\leq k\leq 2^{\lambda}-1$ and $b(x)\in A_{i}/\langle f_{i}(x)^{2^{\lambda}-k-1}\rangle$. In this case, $C_{i}$ satisfies Condition (6) if and only if there exists $u(x)\in A_{i}$ such that

[TABLE]

This implies $f_{i}(x^{-1})^{k}=u(x)f_{i}(x)^{k+1}b(x)$. From this and by $f_{i}(x^{-1})^{k}=x^{-kd_{i}}f_{i}(x)^{k}$, we deduce that $f_{i}(x)^{k}=f_{i}(x)^{k+1}x^{kd_{i}}u(x)b(x)$. This implies that

[TABLE]

and we get a contradiction since $2^{\lambda}-k\geq 1$. Hence $C_{i}$ does not satisfy Condition (6).

Case 5 G=\left(\begin{array}[]{cc}f_{i}(x)^{k}&0\cr 0&f_{i}(x)^{k}\end{array}\right), where $0\leq k\leq 2^{\lambda}$. In this case, we have that $(0,f_{i}(x^{-1})^{k})=x^{4m-kd_{i}}(0,f_{i}(x)^{k})\in C_{i}$ and $(f_{i}(x^{-1})^{k},0)=x^{4m-kd_{i}}(f_{i}(x)^{k},0)\in C_{i}$ by Lemma V.4(ii). Hence $C_{i}$ satisfies Condition (6).

Moreover, by $\|(f_{i}(x)^{k},0)\|_{f_{i}(x)}=\|(0,f_{i}(x)^{k})\|_{f_{i}(x)}=k$ and Lemma V.3(ii), we get $|C_{i}|=2^{(2^{\lambda+1}-k-k)d_{i}}=4^{(2^{\lambda}-k)d_{i}}$.

Cases 6–7 G=\left(\begin{array}[]{cc}f_{i}(x)^{k}&f_{i}(x)^{k}c(x)\cr 0&f_{i}(x)^{k+j}\end{array}\right), where $c(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$, $1\leq j\leq 2^{\lambda}-k-1$ and $0\leq k\leq 2^{\lambda}-2$. By Lemma V.4(ii), it follows that

[TABLE]

This implies $(f_{i}(x^{-1})^{k+j},0)\in C_{i}$. So $C_{i}$ satisfies Condition (6) if and only if there exist $a(x),b(x)\in A_{i}$ such that

[TABLE]

which is equivalent to $f_{i}(x^{-1})^{k}c(x^{-1})=a(x)f_{i}(x)^{k}$ and $f_{i}(x^{-1})^{k}=a(x)f_{i}(x)^{k}c(x)+b(x)f_{i}(x)^{k+j}$. By $f_{i}(x^{-1})^{k}=x^{-kd_{i}}$ in $A_{i}$, the latter condition is equvalent to that

[TABLE]

i.e., $f_{i}(x)^{k}=f_{i}(x)^{k}(c(x)c(x^{-1})+x^{kd_{i}}b(x)f_{i}(x)^{j})$. This condition is equivalent to

[TABLE]

As $1\leq j\leq 2^{\lambda}-k-1$, we conclude that $C_{i}$ satisfies Condition (6) if and only if $c(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$ satisfying $c(x)c(x^{-1})\equiv 1$ (mod $f_{i}(x)^{j}$), i.e., $c(x)\in\mathcal{W}_{i}^{(j)}.$

Conversely, if $c(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$ satisfying $c(x)c(x^{-1})\equiv 1$ (mod $f_{i}(x)^{j}$), then there exists element $g(x)\in A_{i}$ such that $1=c(x)c(x^{-1})+g(x)f_{i}(x)^{j}$. Let $b(x)=x^{-kd_{i}}g(x)\in A_{i}$. Then we have $1=c(x)c(x^{-1})+x^{kd_{i}}b(x)f_{i}(x)^{j}$. This implies that $f_{i}(x)^{k}=f_{i}(x)^{k}c(x^{-1})c(x)+x^{kd_{i}}b(x)f_{i}(x)^{k+j}$. From this and by $f_{i}(x)^{k}=x^{kd_{i}}f_{i}(x^{-1})^{k}$, we deduce that $f_{i}(x^{-1})^{k}=f_{i}(x^{-1})^{k}c(x^{-1})c(x)+b(x)f_{i}(x)^{k+j}$. Select $a(x)=x^{-kd_{i}}c(x^{-1})\in A_{i}$. Then

[TABLE]

and $f_{i}(x^{-1})^{k}=a(x)f_{i}(x)^{k}c(x)+b(x)f_{i}(x)^{k+j}$. Hence

[TABLE]

as required.

Then by Lemma V.3(ii), $\|(f_{i}(x)^{k},f_{i}(x)^{k}c(x))\|_{f_{i}(x)}=k$ and $\|(0,f_{i}(x)^{k+j})\|_{f_{i}(x)}=k+j$, it follows that $|C_{i}|=2^{(2^{\lambda+1}-(k+j)-k)d_{i}}=2^{(2^{\lambda+1}-2k-j)d_{i}}$.

Cases 8–9 G=\left(\begin{array}[]{cc}f_{i}(x)^{k}c(x)&f_{i}(x)^{k}\cr f_{i}(x)^{k+j}&0\end{array}\right), where $c(x)\in f_{i}(x)(A_{i}/\langle f_{i}(x)^{j}\rangle)$, $1\leq j\leq 2^{\lambda}-k-1$ and $0\leq k\leq 2^{\lambda}-2$. Then there exists $g(x)\in A_{i}/\langle f_{i}(x)^{j}\rangle$ such that $c(x)=f_{i}(x)g(x)$. This implies $(f_{i}(x)^{k+1}g(x),f_{i}(x)^{k})\in C_{i}$. Suppose that $(f_{i}(x^{-1})^{k},f_{i}(x^{-1})^{k+1}g(x^{-1}))\in C_{i}$. Then there exist $a(x),b(x)\in A_{i}$ such that

[TABLE]

Hence

[TABLE]

From this and by $f_{i}(x^{-1})^{k}=x^{-kd_{i}}f_{i}(x)^{k}$, we deduce that

[TABLE]

(mod $f_{i}(x)^{2^{\lambda}-k}$), where $2^{\lambda}-k\geq 2$. But $f_{i}(x)$ is nilpotent (mod $f_{i}(x)^{2^{\lambda}-k}$), we get a contradiction. So $C_{i}$ does not satisfy Condition (6) in Theorem V.1.

Summarizing the above, the number of linear codes over $A_{i}$ of length $2$ satisfying Condition (6) in Theorem V.1 (i) is

[TABLE]

After simplifying, we get

[TABLE]

Appendix B: Results for $\mathcal{W}_{1}^{(s)}$ ($1\leq s\leq 4$)

Let $f_{1}(x)=x^{2}+x+1$. For $\mathcal{W}_{1}^{(s)}$, we have the following calculation results:

$\mathcal{W}_{1}^{(1)}=\{1,x,1+x\}$;

$\mathcal{W}_{1}^{(2)}=\{1,1+(1+x)f_{1}(x),x,x+f_{1}(x),1+x+f_{1}(x),1+x+(1+x)f_{1}(x)\}$;

$\mathcal{W}_{1}^{(3)}=\{1,x^{5}+x^{3}+x+1,x^{3},x^{5}+x,x,x^{5}+x^{4}+x^{3}+x^{2}+1,x^{4},x^{5}+x^{3}+x^{2}+x+1,x^{5},x^{5}+x^{4}+x^{2}+1,x^{2},x^{4}+1\}$;

$\mathcal{W}_{1}^{(4)}=\{1,x^{6}+x^{5}+x^{3}+x,x^{5}+x^{3}+x+1,x^{6},x^{3},x^{6}+x^{5}+x+1,x^{5}+x,x^{6}+x^{3}+1,x,x^{7}+x^{6}+x^{4}+x^{2},x^{7},x^{6}+x^{4}+x^{2}+x,x^{4},x^{7}+x^{6}+x^{2}+x,x^{7}+x^{4}+x,x^{6}+x^{2},x^{5},x^{7}+x^{4}+x^{3}+x^{2}+1,x^{5}+x^{4}+x^{2}+1,x^{7}+x^{3},x^{2},x^{7}+x^{5}+x^{4}+x^{3}+1,x^{4}+1,x^{7}+x^{5}+x^{3}+x^{2}\}$.

Appendix C: Expressions for $a(x)$ and $b(x)$ in Example IV.3

Case 1 $a(x)=x^{4}+x^{2}+1$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$1001011100010100$, $1000001101010001$, $1001001101010000$,

$1000011100010101$, $0100001000001101$, $1001011001001000$,

$0100011001001001$, $0101001000001100$, $0001111001010110$,

$0100111101000010$, $0000111101000110$, $0101111001010010$,

$1000001110010111$, $1101001010000011$, $1001001010000111$,

$1100001110010011$, $0000110001011110$, $0100100100001111$,

$0100100000011110$, $0000110101001111$, $0001010010000011$,

$0101000111010010$, $0101000011000011$, $0001010110010010$.

Case 2 $a(x)=x^{6}+x^{4}+x^{2}$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$1001011100010100$, $1000001101010001$, $1001001101010000$,

$1000011100010101$, $1111010001100000$, $1110000000100101$,

$1111000000100100$, $1110010001100001$, $1100010111100000$,

$1001010011110100$, $1101010011110000$, $1000010111100100$,

$1000001110010111$, $1101001010000011$, $1001001010000111$,

$1100001110010011$, $0110000110000101$, $0010010011010100$,

$0010010111000101$, $0110000010010100$, $0001010010000011$,

$0101000111010010$, $0101000011000011$, $0001010110010010$.

Case 3 $a(x)=x^{6}+x^{4}+1$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$0010000101111001$, $0011010100111100$, $0010010100111101$,

$0011000101111000$, $1111010001100000$, $1110000000100101$,

$1111000000100100$, $1110010001100001$, $1100010111100000$,

$1001010011110100$, $1101010011110000$, $1000010111100100$,

$0101100000100001$, $0000100100110101$, $0100100100110001$,

$0001100000100101$, $0110000110000101$, $0010010011010100$,

$0010010111000101$, $0110000010010100$, $0111100101011000$,

$0011110000001001$, $0011110100011000$, $0111100001001001$.

Case 4 $a(x)=x^{6}+x^{2}+1$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$0010000101111001$, $0011010100111100$, $0010010100111101$,

$0011000101111000$, $0100001000001101$, $0101011001001000$,

$0100011001001001$, $0101001000001100$, $0001111001010110$,

$0100111101000010$, $0000111010001100$, $0101111001010010$,

$0101100000100001$, $0000100100110101$, $0100100100110001$,

$0001100000100101$, $0000110001011110$, $0100100100001111$,

$0100100000011110$, $0000110101001111$, $0111100101011000$,

$0011110000001001$, $0011110100011000$, $0111100001001001$.

Case 5 $a(x)=x^{5}+x^{3}+x$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$0010100011101001$, $1000101011000001$, $0000101011001001$,

$1010100011100001$, $0000011000101111$, $1010010000000111$,

$0010010000001111$, $1000011000100111$, $0100101110001010$,

$1100000100101000$, $1100001100001010$, $0100100110101000$,

$1010000110000110$, $0010101100100100$, $0010100100000110$,

$1010001110100100$, $0000011110100011$, $0010111100101001$,

$0000111100101011$, $0010011110100001$, $1100000101001011$,

$1110100111000001$, $1100100111000011$, $1110000101001001$.

Case 6 $a(x)=x^{7}+x^{5}+x^{3}$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$0010100011101001$, $1000101011000001$, $0000101011001001$,

$1010100011100001$, $1011000001000010$, $0001001001101010$,

$1001001001100010$, $0011000001001010$, $0100101110001010$,

$1100000100101000$, $1100001100001010$, $0100100110101000$,

$0111101000110000$, $1111000010010010$, $1111001010110000$,

$0111100000010010$, $0110101001111000$, $0100001011110010$,

$0110001011110000$, $0100101001111010$, $1100000101001011$,

$1110100111000001$, $1100100111000011$, $1110000101001001$.

Case 7 $a(x)=x^{7}+x^{5}+x$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$1001111010000100$, $0011110010101100$, $1011110010100100$,

$0001111010001100$, $1011000001000010$, $0001001001101010$,

$1001001001100010$, $0011000001001010$, $1001000000111100$,

$0001101010011110$, $0001100010111100$, $1001001000011110$,

$0111101000110000$, $1111000010010010$, $1111001010110000$,

$0111100000010010$, $0110101001111000$, $0100001011110010$,

$0110001011110000$, $0100101001111010$, $1010110010010000$,

$1000010000011010$, $1010010000011000$, $1000110010010010$.

Case 8 $a(x)=x^{7}+x^{3}+x$ and $b(x)=\sum_{j=0}^{15}b_{j}x^{j}$, where $b_{0}b_{1}b_{2}\ldots b_{15}\in\mathbb{F}_{2}^{16}$ is given by one of the following $24$ cases:

$1001111010000100$, $0011110010101100$, $1011110010100100$,

$0001111010001100$, $0000011000101111$, $1010010000000111$,

$0010010000001111$, $1000011000100111$, $1001000000111100$,

$0001101010011110$, $0001100010111100$, $1001001000011110$,

$1010000110000110$, $0010101100100100$, $0010100100000110$,

$1010001110100100$, $0000011110100011$, $0010111100101001$,

$0000111100101011$, $0010011110100001$, $1010110010010000$,

$1000010000011010$, $1010010000011000$, $1000110010010010$.

Appendix D: Expressions for $a(x)$ and $\eta(x)$ in Example IV.4

Case 1 $a(x)\in\{x^{2}+x+1,x^{3}+x^{2}+1,x^{3}+x^{2}+x,x^{3}+x+1\}$ and $\eta(x)=\sum_{j=0}^{11}\eta_{j}x^{j}$, where $\eta_{0}\eta_{1}\eta_{2}\ldots\eta_{11}\in\mathbb{F}_{2}^{12}$ is given by one of the following $168$ cases:

$000000001101$, $000000010111$, $000000011010$, $000000101110$,

$000000110100$, $000000111001$, $000001011100$, $000001100101$,

$000001101000$, $000001110010$, $000010011011$, $000010111000$,

$000011001010$, $000011010000$, $000011011101$, $000011100100$,

$000100110110$, $000100111011$, $000101001001$, $000101010011$,

$000101100111$, $000101110000$, $000110000011$, $000110010100$,

$000110100000$, $000110111010$, $000111000101$, $000111001000$,

$001000001001$, $001000100111$, $001001010101$, $001001100001$,

$001001101100$, $001001110110$, $001010010010$, $001010100110$,

$001010101011$, $001010110001$, $001011001110$, $001011100000$,

$001100000110$, $001100001011$, $001100100101$, $001100101000$,

$001101000000$, $001101010111$, $001101100011$, $001101110100$,

$001110001010$, $001110010000$, $001110101001$, $001110110011$,

$010000010010$, $010000110001$, $010001000011$, $010001001110$,

$010001011001$, $010001110111$, $010010001001$, $010010101010$,

$010011000010$, $010011010101$, $010011011000$, $010011101100$,

$010100000111$, $010100011101$, $010100100100$, $010100110011$,

$010101001100$, $010101010110$, $010101100010$, $010101110101$,

$010110010001$, $010110011100$, $010111000000$, $010111001101$,

$011000000001$, $011000001100$, $011000010110$, $011000110101$,

$011001000111$, $011001001010$, $011001010000$, $011001110011$,

$011010000000$, $011010101110$, $011011000110$, $011011101000$,

$011100000011$, $011100010100$, $011100100000$, $011100110111$,

$011101000101$, $011101010010$, $011101100110$, $011101110001$,

$100000011101$, $100000100100$, $100000101001$, $100000110011$,

$100001000001$, $100001100010$, $100010000110$, $100010001011$,

$100010011100$, $100010110010$, $100011001101$, $100011101110$,

$100100000101$, $100100010010$, $100100101011$, $100100110001$,

$100101010100$, $100101011001$, $100110000100$, $100110010011$,

$100110101010$, $100110110000$, $100111010101$, $100111011000$,

$101000000011$, $101000001110$, $101000011001$, $101000111010$,

$101001001000$, $101001100110$, $101010011000$, $101010100001$,

$101010101100$, $101010111011$, $101011000100$, $101011101010$,

$101100000001$, $101100011011$, $101100100010$, $101100111000$,

$101110000000$, $101110011010$, $101110100011$, $101110111001$,

$110000000010$, $110000010101$, $110000011000$, $110000101100$,

$110001001001$, $110001101010$, $110010001110$, $110010010100$,

$110010011001$, $110010100000$, $110011000101$, $110011100110$,

$110100000000$, $110100001101$, $110101010001$, $110101011100$,

$110110000001$, $110110001100$, $110111010000$, $110111011101$,

$111000000110$, $111000101000$, $111001000000$, $111001101110$,

$111010001010$, $111010100100$, $111011001100$, $111011100010$.

Case 2 $a(x)=x^{2}+x+1$ and $\eta(x)=\sum_{j=0}^{11}\eta_{j}x^{j}$, where $\eta_{0}\eta_{1}\eta_{2}\ldots\eta_{11}\in\mathbb{F}_{2}^{12}$ is given by one of the following $14$ cases:

$000001000110$, $000010001100$, $001000000100$, $001001000010$,

$010000001000$, $010000100110$, $010001100000$, $010010000100$,

$011000100010$, $011001100100$, $100001001100$, $100011000000$,

$110001000100$, $110011001000$.

Case 3 $a(x)=x^{3}+x^{2}+1$ and $\eta(x)=\sum_{j=0}^{11}\eta_{j}x^{j}$, where $\eta_{0}\eta_{1}\eta_{2}\ldots\eta_{11}\in\mathbb{F}_{2}^{12}$ is given by one of the following $14$ cases:

$000000100011$, $000010000001$, $000100000010$, $000100011000$,

$000100100001$, $000110011001$, $001000010011$, $001000110000$,

$001100010001$, $001100110010$, $100000010000$, $100010010001$,

$100100001000$, $100110001001$.

Case 4 $a(x)=x^{3}+x^{2}+x$ and $\eta(x)=\sum_{j=0}^{11}\eta_{j}x^{j}$, where $\eta_{0}\eta_{1}\eta_{2}\ldots\eta_{11}\in\mathbb{F}_{2}^{12}$ is given by one of the following $14$ cases:

$000000100011$, $000001000110$, $000100000010$, $000100100001$,

$001000000100$, $001000010011$, $001000110000$, $001001000010$,

$001100010001$, $001100110010$, $010000100110$, $010001100000$,

$011000100010$, $011001100100$.

Case 5 $a(x)=x^{3}+x+1$ and $\eta(x)=\sum_{j=0}^{11}\eta_{j}x^{j}$, where $\eta_{0}\eta_{1}\eta_{2}\ldots\eta_{11}\in\mathbb{F}_{2}^{12}$ is given by one of the following $14$ cases:

$000010000001$, $000010001100$, $000100011000$, $000110011001$,

$010000001000$, $010010000100$, $100000010000$, $100001001100$,

$100010010001$, $100011000000$, $100100001000$, $100110001001$,

$110001000100$, $110011001000$.