# Self-dual binary [8m, 4m]-codes constructed by left ideals of the dihedral group algebra F_2[D_8m]

Let m be an arbitrary positive integer and D_8m be a dihedral group of order 8m, i.e., D_8m=〈 x,y| x^4m=1, y^2=1, yxy=x^-1〉. Left ideals of the dihedral group algebra 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 binary self-dual codes of length a multiple of 8. Moreover, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing 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 192 extremal self-dual binary [48,24,12]-codes and 728 extremal self-dual binary [56,28,12]-codes.

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## 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 (see [1], for an example).

A linear code is said to be self-dual if . Binary self-dual codes are called Type II if the weights of all codewords are multiple of 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 of a binary self-dual code of length is

 d≤{4⌊n24⌋+6,if n≡22 (mod 24);4⌊n24⌋+4,otherwise.

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

Let be a finite field of elements and be a finite group. The group algebra is an -algebra with basis . Addition, multiplication with scalars and multiplication are defined by:

 ∑g∈Gagg+∑g∈Gbgg=∑g∈G(ag+bg)g, c(∑g∈Gagg)=∑g∈Gcagg,
 (∑g∈Gagg)(∑g∈Gbgg)=∑g∈G(∑uv=gaubv)g,

for any and . Then is a noncommutative ring with identity where and is the identity elements of and respectively. It is known that is semisimple if and only if .

In this paper, let

 D2n = ⟨x,y∣xn=1,y2=1,yxy=x−1⟩ = {xiyj∣0≤i≤n−1,j=0,1}

be a dihedral group of order . For any , , we define

 Ψ(a)=n−1∑i=0ai,0xi+n−1∑i=0ai,1xiy.

Then is an isomorphism of -linear spaces from onto . As a natural generalization of Dutra et al. [2], a nonempty subset of the -linear space is called a left dihedral code (or left -code for more clear) over if is a left ideal of . We will equate with 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 [2] investigated codes that are two-sided ideals in a semisimple finite group algebra , and given a criterion to decide if these ideals are all the minimal two-sided ideals of when is a dihedral group. Brochero Martínez [3] shown all central irreducible idempotents and their Wedderburn decomposition of the semisimple dihedral group algebra when every divisor of divides . Moreover, we gave a system theory for left -codes over finite fields in [4] where , and obtained a complete description for left -codes over Galois rings in [5] were .

McLoughlin [6] provided a new 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 . Dougherty et al. [7] and [1] gave constructions of self-dual and formally self-dual codes from group rings where the ring 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 shown precisely which groups can be used to construct the extremal Type II codes of length 24 and 48.

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 . There are still many problems worth studying in this field. For examples,

For which does there exists a doubly-even self-dual binary code (Open Question 7.7 in [8])?

Now, we study binary self-dual codes in a new way different from the methods used in [6], [7] and [1]. We give an explicit construction and enumeration for all distinct self-dual binary left -codes in this paper. These codes make up an important class of binary self-dual codes of length a multiple of . In future work, we will try to determine extremal doubly-even binary self-dual codes of length a multiple of among these codes.

###### Notation I.1

In this paper, let be a binary field and , where and are nonnegative integers such that

is odd. Then

where

 4m=2λ⋅m0 and λ=λ0+2≥2.

The present paper is organized as follows. In section 2, we give a concatenated structure of any binary left -code. Precisely, we show that every binary left -code can be decomposed into a direct sum of concatenated codes, where the inner code is a binary cyclic code of length and the outer code is a special linear code over finite commutative chain rings of length . In Section 3, we provide an explicit representation and enumeration for all distinct binary left -codes. Specifically, we give a unique generator matrix of the outer code for each concatenated code in the decomposition of any binary left -code. In Section 4, we determine the dual code for each code and give an explicit representation for all distinct self-dual binary left -codes. In Section 5, we give recursive algorithms to solve problems in the construction of self-dual binary left -codes and obtain a clear formula to count the number of all these self-dual codes. In Section 6, we list all distinct self-dual binary left -codes for , and provide extremal self-dual binary codes with parameters , , , , respectively. Section 7 concludes the paper.

## Ii Concatenated structure of left D8m-codes

In this section, we introduce the necessary notations and sketch the results of finite commutative chain rings first. Then we give a concatenated structure of any binary left -code.

For any nonzero polynomial of degree , the reciprocal polynomial of is defined by

and is said to be self-reciprocal if .

As is an odd positive integer, we have that

 xm0−1=r∏i=0fi(x),

where are pairwise coprime irreducible polynomials in such that

• There are nonnegative integers and such that

 r=ρ+2ϵ.
• with degree .

• is self-reciprocal and of degree for all .

• is not self-reciprocal, and for all .

It is clear that and

 x4m−1=r∏i=0fi(x)2λ.

This implies . In this paper, we denote

where we regard elements of as polynomials in of degree and the arithmetic is done modulo .

where we regard elements of as polynomials in of degree and the arithmetic is done modulo , for all .

The rings () are 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 for any and . Then we have the following conclusions.

(i) is a finite chain ring, is the unique maximal ideal of , the nilpotency index of is and .

(ii) Let . Then every element of has a unique -expansion:

 2λ−1∑j=0aj(x)fi(x)j, where aj(x)∈Ti, ∀j=0,1,…,2λ−1.

(iii) We can regard elements of as polynomials in of degree and the arithmetic is done modulo . Hence .

For each , denote . Then and are coprime polynomials. Hence there are polynomials such that

 ui(x)Fi(x)+vi(x)fi(x)=1.

This implies and

 ui(x2λ)Fi(x2λ)+vi(x2λ)fi(x)2λ=1. (1)

In the rest of this paper, let satisfying

 εi(x)≡ui(x2λ)Fi(x2λ) (mod x4m−1). (2)

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

###### Lemma II.2

(i) , and for all in the ring .

(ii) , where with as its multiplicative identity. Moreover, this decomposition is a ring direct sum in that for all .

(iii) For each , the map

 φi:a(x)↦εi(x)a(x) (mod x4m−1), ∀a(x)∈Ai

is an isomorphism of rings from onto .

(iv) For any , , define

 φ:(a0(x),…,ar(x))↦r∑i=0εi(x)ai(x) (mod x4m−1).

Then is a ring isomorphism from the direct product ring onto .

As usual, we equate each vector

with . Then binary cyclic codes of length are identified with ideals of the ring . In particular, we have the following properties for the ideal of .

###### Corollary II.3

Let . Then

(i) is a binary cyclic code of length with parity check polynomial and generating idempotent .

(ii) As a binary linear code of length ,

 {εi(x),xεi(x),x2εi(x),…,x2λdi−1εi(x)}

is a basis of . Hence .

###### Proof:

(ii) Since is an -linear space with a basis , by Lemma II.2(iii) we see that , is an -basis of .

Now, let be a linear code of length over , i.e. is an -submodule of . For each , we denote by

 w(Ai)H(ξ)=|{j∣bj(x)≠0 in Ai, j=0,1}|

the Hamming weight of and define the minimum Hamming distance of as

 d(Ai)H(Ci)=min{w(Ai)H(ξ)∣ξ≠0, ξ∈Ci}.

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 of the inner code and the outer code by

 Ai□φiCi = {(φi(ξ0),φi(ξ1))∣(ξ0,ξ1)∈Ci} = {(εi(x)ξ0,εi(x)ξ1)∣(ξ0,ξ1)∈Ci} ⊆ A2i.

By Lemma II.2(iii), we conclude that is a binary quasi-cyclic code of length and index and the number of codewords is equal to . This implies

 dimF2(Ai□φiCi)=log2|Ci|,

and the minimum Hamming distance of satisfies

 d(F2)H(Ai□φiCi)≥d(F2)H(Ai)⋅d(Ai)H(Ci),

where is the minimum Hamming weight of as a binary linear code of length .

As the end of this section, we consider the structure of binary left -codes. As

 D8m=⟨x,y∣x4m=1,y2=1,yxy=x−1⟩,

is a cyclic subgroup of with order generated by . It is obvious that the group algebra is equal to the residue class ring . Hence is a subring of and

 F2[D8m]={α(x)+β(x)y∣α(x),β(x)∈A} (y2=1)

in which , . Now, we define a map by

 Θ(α(x),β(x))=α(x)+β(x)y, ∀α(x),β(x)∈A.

Then one can easily verify that is an -module isomorphism from onto .

Let be a nonempty subset of . Then is a left ideal of , i.e. is a binary left -code, if and only if is an -submodule of and for any . From this and by

 yξ=α(x−1)y+β(x−1)y2=β(x−1)+α(x−1)y,

we deduce that

 yξ∈C⟺(β(x−1),α(x−1))∈Θ−1(C).

Let . Then . Hence is a binary left -code if and only if there is a unique -submodule of satisfying the following condition:

 (β(x−1),α(x−1))∈C′, ∀(α(x),β(x))∈C′

such that . We will equate with in this paper.

###### Theorem II.4

Every binary left -code can be uniquely decomposed as the following:

 C=r⨁i=0(Ai□φ1Ci)=r∑i=0(Ai□φ1Ci),

where , , is a linear code of length over the finite chain ring satisfying the following condition

(i) If , satisfies

 (bi(x−1),ai(x−1))∈Ci, ∀(ai(x),bi(x))∈Ci. (3)

(ii) If , the pair of linear codes is given by

 Ci+ϵ={(bi(x−1),ai(x−1))∣(ai(x),bi(x))∈Ci}⊆A2i+ϵ,

where is an arbitrary linear code of length over .

Moreover, the number of codewords in is .

###### Proof:

For any integer , , denote

 μ(i)=⎧⎪⎨⎪⎩i,when 0≤i≤ρ;i+ϵ,when ρ+1≤i≤ρ+ϵ;i−ϵ,when ρ+ϵ+1≤i≤ρ+2ϵ.

Now, we claim that

 εi(x−1)=εμ(i)(x) in A, where x−1=x4m−1. (4)

In fact, by we have

 fi(x−1)=x−difμ(i)(x) in A

and . From these, we deduce that and

 Fi(x−2λ) = x−4m−1fi(x−1)2λ=x4m+1x4mfi(x−1)2λ = x4m+1x4m−2λdifμ(i)(x)2λ = x2λdi−4mFμ(i)(x).

By Equation (1), it follows that

 ui(x−2λ)Fi(x−2λ)+vi(x−2λ)fi(x−1)2λ=1,

where

 vi(x−2λ)fi(x−1)2λ = vi(x−2λ)x−2λdifμ(i)(x)2λ = ˆvμ(i)(x)fμ(i)(x)2λ,
 ui(x−2λ)Fi(x−2λ) = ui(x−2λ)x2λdi−4mFμ(i)(x2λ) = ˆuμ(i)(x)Fμ(i)(x2λ),

and satisfying:

;

,

respectively. Hence

 ˆuμ(i)(x)Fμ(i)(x2λ)+ˆvμ(i)(x)fμ(i)(x)2λ=1.

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

 εi(x−1) = ui(x−2λ)Fi(x−2λ)=ˆuμ(i)(x)Fμ(i)(x2λ) = εμ(i)(x).

Therefore, Equation (4) was proved.

Using the notations in Lemma II.2(iii) and (iv), for any , , we define

 Φ((ξ00,ξ01),(ξ10,ξ11),…,(ξr0,ξr1)) = r+t∑i=0(φi(ξi0),φi(ξi1)) = (r∑i=0εi(x)ξi0,r∑i=0εi(x)ξi1) (mod x4m−1).

It is clear that is an -module isomorphism from onto . Now, let be an -submodule of . Then for each integer , , there is a unique -submodule of such that

 C = Φ(C0×C1×…×Cr) = {r∑i=0(φi(ξi0),φi(ξi1))∣(ξi0,ξi1)∈Ci, 0≤i≤r} = r⨁i=0{(φi(ξi0),φi(ξi1))∣(ξi0,ξi1)∈Ci} = (A0□φ0C0)⊕(A1□φ1C1)⊕…⊕(Ar□φrCr).

It is obvious that .

Moreover, for any integer , , and , let where

 α(x)=r∑i=0εi(x)ai(x) and β(x)=r∑i=0εi(x)bi(x).

By Equation (4), in the ring we have

;

.

These imply

 (β(x−1),α(x−1)) = (r∑i=0εμ(i)(x)bi(x−1),r∑i=0εμ(i)(x)ai(x−1)) = Φ((b0(x−1),a0(x−1)),…,(bρ(x−1),aρ(x−1)), (bρ+ϵ+1(x−1),aρ+ϵ+1(x−1)), …,(bρ+2ϵ(x−1),aρ+2ϵ(x−1)), (bρ+1(x−1),aρ+1(x−1)), …,(bρ+ϵ(x−1),aρ+ϵ(x−1))).

From this and by , we deduce that

 (β(x−1),α(x−1))∈C⟺(bi(x−1),ai(x−1))∈Cμ(i), ∀i.

Therefore, we have one of the following two cases:

(i) Let . In this case, we have and hence satisfies Condition (3).

(ii) Let . we have and . In this case, and satisfy the above conditions if and only if and is an arbitrary linear code over of length .

In the rest of this paper, we call the canonical form decomposition of the binary left -code .

By Theorem II.4, in order to list all distinct binary left -codes we just need to solve the following questions:

Question 1. Determine all linear codes of length over for each .

Question 2. Determine all linear codes of length over satisfying Condition (3) for each .

We will solve these two questions in the next section.

## Iii Representation and enumeration for binary left D8m-codes

In this section, we solve the two questions at the end of Section II first. Then we obtain an explicit representation and enumeration for binary left -codes.

Let and . By Lemma II.1(ii), has a unique -adic expansion:

 a(x)=2λ−1∑j=0fi(x)jaj(x), aj(x)∈Ti.

If , we define the -degree of as the least index for which and denote as