 # Explicit representation for a class of Type 2 constacyclic codes over the ring F_2^m[u]/〈 u^2λ〉 with even length

Let F_2^m be a finite field of cardinality 2^m, λ and k be integers satisfying λ,k≥ 2 and denote R=F_2^m[u]/〈 u^2λ〉. Let δ,α∈F_2^m^×. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ+α u^2)-constacyclic codes over R of length 2^kn, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every (δ+α u^2)-constacyclic code over R of length 2^kn is an ideal generated by at most 2 polynomials in the residue class ring R[x]/〈 x^2^kn-(δ+α u^2)〉.

## Authors

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Introduction

Algebraic coding theory deals with the design of error-correcting and error-detecting codes for the reliable transmission of information across noisy channel. The class of constacyclic codes plays a very significant role in the theory of error-correcting codes. It includes as a subclass of the important class of cyclic codes, which has been well studied since the late 1950’s. Constacyclic codes also have practical applications as they can be efficiently encoded with simple shift registers. This family of codes is thus interesting for both theoretical and practical reasons.

Let be a commutative finite ring with identity , and be the multiplicative group of invertible elements of . For any , we denote by , or for simplicity, the ideal of generated by , i.e. . For any ideal of , we will identify the element of the residue class ring with (mod ) for any in this paper.

A code over of length is a nonempty subset of , . The code is said to be linear if is an -submodule of . All codes in this paper are assumed to be linear. The ambient space is equipped with the usual Euclidian inner product, i.e. , where , and the dual code is defined by . If , is called a self-dual code over . Let . Then a linear code over of length is called a -constacyclic code if for all . Particularly, is called a negacyclic code if , and is called a cyclic code if .

For any , let . We will identify with in this paper. Then is a -constacyclic code over of length if and only if is an ideal of the residue class ring , and the dual code of is a -constacyclic code of length over , i.e. is an ideal of the ring (cf.  Propositions 2.4 and 2.5). The ring is usually called the ambient ring of -constacyclic codes over with length . In addition, is called a simple-root constacyclic code if , and called a repeated-root constacyclic code otherwise.

Let be a finite field of cardinality , where is power of a prime, and denote () where . Then is a finite chain ring. As in Dinh et al , if

 γ=α0+αkuk+…+αe−1ue−1

where satisfying , then is called a unit in to be of Type . Especially, is called a unit in to be of Type if . When is a unit in of Type , a -constacyclic code of length over is said to be of Type . Especially, cyclic codes and negacyclic codes are both constacyclic codes over of Type .

For examples, let and . Then is a a unit in of Type 2. Hence -constacyclic codes form a typical subclass of the class of all Type 2 constacyclic codes over the finite chain ring .

When , there were a lot of literatures on linear codes, Type constacyclic codes and some special class of Type constacyclic codes of length over rings for various prime and positive integers and . See , , [13–19], ,  and , for examples. In particular, we  gave an explicit representation and a complete classification for all Type repeated-root constacyclic codes over and their dual codes for any prime number and positive integer .

When , the structures for repeated-root constacyclic codes of Type 1 over had been studied by many literatures. For examples, Kai et al.  investigated -constacyclic codes of arbitrary length over , where . Cao  generalized these results to -constacyclic codes of arbitrary length over an arbitrary finite chain ring , where is a unit of and generates the unique maximal ideal of with nilpotency index . Hence every Type 1 constacyclic code over any finite chain ring is a one-generator ideal of the ambient ring.

When , we known the following literatures for the structures of -constacyclic codes over , where :

Let .

Sobhani  determined the structure of -constacyclic codes of length over .

On the basis of , using methods different from  we gave a complete description for -constacyclic codes over the ring of length and determine explicitly the self-dual codes among them for any odd positive integer .

Let .

When , in  for any , an explicit representation for all distinct -constacyclic codes over the ring of length is given, and the dual code for each of these codes is determined. For the case of and , all self-dual -constacyclic codes over of length are provided.

When , in  an explicit representation for all distinct -constacyclic codes over of length was given, where . Formulas for the number of all such codes and the number of codewords in each code are provided respectively, and the dual code for each of these codes was determined explicitly.

When , in  a representation and enumeration formulas for all distinct -constacyclic codes over of length were presented explicitly, where is odd.

Mahmoodi and Sobhani  gave a complete classification for -constacyclic codes of length over , where . They determined self-dual such codes and enumerate them for the case . Moreover, the authors discussed on Gray-maps on which preserve self-duality, and also discuss on the images of self-dual constacyclic codes under these Gray maps.

Let , where is an arbitrary integer such that .

Based on the results of , we gave a complete description for repeated-root -constacyclic codes over for any odd prime . The expressions and their derivation process for the main results in  are heavily depend on that is an odd prime.

Many methods and techniques used in  and  can not be directly applied to the case .

Motivated by those, we follow the main idea in , promote and develop the methods used in  to determine all -constacyclic codes over of arbitrary even length. The ideas and methods used in this paper are different to that used in  and . Therefore, we can come to clearer and more precise conclusions:

Provide an explicit representation and enumeration for all distinct -constacyclic codes over of length through only one theorem, for any integer and odd positive integer .
Although the proof of this theorem is somewhat complicated, the results expressed by the theorem are very clear and direct.

Obtain a clear and exact formula to count the number of all -constacyclic codes over of length , and give a clear formula to count the number of codewords in each code from its generators directly.

The present paper is organized as follows. In Section 2, we provide the notations and review preparation results necessary. Then we give the main result (Theorem 2.5) for representation and enumeration for all distinct -constacyclic code over of length . In Section 3, we give an explicit representation for a special subclass of -constacyclic code over of length including the particular situation of . Based on this, we correct a mistake for -constacyclic codes of length over listed by an example of . Moreover, we list precisely all distinct self-dual -constacyclic codes of length over for any . In Section 4, we give a proof for the main result in Section 2. Section 5 concludes the paper.

## 2 Main results

In this section, we introduce the necessary notations and review preparation results first. Then we provide the main result on representation and enumeration for -constacyclic codes over the ring of length where .

In this paper, we always assume that are positive integers such that is odd and . As and , there exists a unique element such that . This implies in . In this paper, we adopt the following notation.

• (). Then is a finite chain ring with the unique maximal ideal , and is the nilpotent index of .

• in which the arithmetic is done modulo . Then is a finite principal ideal ring and .

• in which the operations are defined by

• ,

• ,

for all . Then is a subring of .

It is clear that both and are -spaces of dimension . Precisely, we have that

is an -basis of ;

is an -basis of .

Hence there is a unique isomorphism of -spaces from onto such that

 Ψ(xi(x2kn)l)=xi(δ+αu2)l and Ψ(uxi(x2kn)l)=uxi(δ+αu2)l, (1)

for all integers and satisfying and respectively. Moreover, by it follows that

In the ring , we have and . These imply and , respectively.

In the ring , we have and . These imply .

From these and by an argument similar to the proof of Theorem 2.1 in , one can easily verify the following conclusion.

Lemma 2.1 Using the notations above, is a ring isomorphism from onto .

From this lemma and by Equation (1), we deduce the following:

 Ψ((xn+δ0)i+l⋅2kn)=αlu2l(xn+δ0)i, ∀0≤i≤2kn−1, ∀0≤l≤λ−1. (2)

In the rest of this paper, we usually identify with under the ring isomorphism determined by Equations (1) and (2), unless otherwise stated. Hence in order to determine all -constacyclic codes over of length , it is sufficient to give all ideals of the ring .

To determine all ideals of the ring , we need to study the structure of the ring first. As is odd, there are pairwise coprime monic irreducible polynomials in such that . This implies

 (x2kn−δ)λ=(xn+δ0)2kλ=f1(x)2kλ…fr(x)2kλ.

For any integer , , we assume and denote . Then and . These imply

 (x2kn−δ)λ=Fj(x)2kλfj(x)2kλ and gcd(Fj(x)2kλ,fj(x)2kλ)=1.

Hence there exist such that

 gj(x)Fj(x)λpk+hj(x)fj(x)λpk=1.

As in  for odd prime , we adopt the following notation where be an integer satisfying .

• Let be defined by

 εj(x)≡gj(x)Fj(x)2kλ=1−hj(x)fj(x)2kλ (mod (xn+δ0)2kλ).
• in which the arithmetics are done modulo .

• in which the arithmetics are done modulo . Then is an extension field of with elements.

Remark is a finite field in which the arithmetic is done modulo , is a finite ring in which the arithmetic is done modulo and is a principal ideal ring in which the arithmetic is done modulo . In this paper, we adopt the following points of view:

 Fj⊆Kj⊆A as sets.

It is worth noting that is not a subfield of and is not a subring of when .

Then from Chinese remainder theorem for commutative rings, we deduce the following lemma about the structure and properties of the ring .

Lemma 2.2 Using the notations above, we have the following decomposition via idempotents:

(i) , and in the ring for all .

(ii) We regard as a subset of for all . Then

 A=ε1(x)K1⊕ε2(x)K2⊕…⊕εr(x)Kr (mod (xn+δ0)2kλ),

where for all .

For the ring , where , we know the following conclusion.

Lemma 2.3 (cf.  Example 2.1) The ring have the following properties:

(i) is a finite chain ring, generates the unique maximal ideal of , is the nilpotency index of and the residue class field of modulo is .

(ii) Every element of has a unique -adic expansion:

 ξ=b0(x)+b1(x)fj(x)+…+b2kλ−1fj(x)2kλ−1,

where . Moreover, is invertible in if and only if . Here regard as a subset of .

(iii) All distinct ideals of are given by:

 ⟨fj(x)l⟩=fj(x)lKj, 0≤l≤2kλ.

Moreover, for .

(iv) Let . Then . Precisely, we have

 Kj/⟨fj(x)l⟩={l−1∑i=0bi(x)f(x)i∣bi(x)∈Fj, i=0,1,…,l−1}.

Remark For any integer , , by Lemma 2.3(iv) we can identify with up to a natural ring isomorphism. We will take this view in the rest of this paper. Then for any , we stipulate

 fj(x)l⋅F2m[x]⟨fj(x)t⟩ = fj(x)l(Kj/⟨fj(x)t⟩) = {t−1∑i=lbi(x)f(x)i∣bi(x)∈Fj, i=l,…,t−1}.

Hence , where we set for convenience.

Then we study the structure of the ring . To do this, we introduce the following notation.

• Let satisfying .

• Let be defined by .

• in which the operations are defined by

• ,

• ,

for all . Then is a subring of .

Lemma 2.4 Let . Then we have the following conclusions.

(i) The element is invertible in the ring and satisfies

 ω2jfj(x)2k≡α−1(xn+δ0)2k (mod  fj(x)2kλ).

Hence in the ring .

(ii) We regard as a subset of for all . Then

 A+uA=r⨁j=1εj(x)(Kj+uKj)=r∑j=1εj(x)(Kj+uKj) (mod (xn+δ0)2kλ),

where .

Proof. (i) Since satisfying (mod ) and , we conclude that as polynomials in . This implies that is an invertible element of the ring . Then from and , we deduce that

 ω2jfj(x)2k ≡ (α0Fj(x)2k−1)2fj(x)2k=α−1Fj(x)2kfj(x)2k ≡ α−1(xn+δ0)2k (mod  fj(x)2kλ).

This implies in .

(ii) By Lemma 2.2 and the conclusion of (i), it follows that

 A+uA = A[u]⟨u2−α−1(xn+δ0)2k⟩=(⨁rj=1εj(x)Kj)[u]⟨u2−α−1(xn+δ0)2k⟩ = r⨁j=1εj(x)⋅Kj[u]⟨u2−α−1(xn+δ0)2k⟩ = r⨁j=1εj(x)⋅Kj[u]⟨u2−ω2jfj(x)2k⟩. (mod (xn+δ0)2kλ).

This implies .

Finally, we list all -constacyclic codes over of length by the following theorem. Its proof will be given in Section 4.

Theorem 2.5 For any integers : and , let

 F2m[x]⟨fj(x)l⟩={l−1∑i=0ai(x)fj(x)i∣a0(x),a1(x),…,al−1(x)∈Fj},

Especially, we have and . Then all distinct -constacyclic codes over of length , as ideals of the ring , are given by

 C=r⨁j=1εj(x)Cj,

where is an ideal of the ring listed by the following three cases.

(I) ideals given by the following two subcases:

1. with , where and .

2. with ,
where and .

(II) ideals:

3. with , where .

(III) ideals given by the following three subcases:

4. with ,
where