 # Matrix-product structure of constacyclic codes over finite chain rings F_p^m[u]/〈 u^e〉

Let m,e be positive integers, p a prime number, F_p^m be a finite field of p^m elements and R=F_p^m[u]/〈 u^e〉 which is a finite chain ring. For any ω∈ R^× and positive integers k, n satisfying gcd(p,n)=1, we prove that any (1+ω u)-constacyclic code of length p^kn over R is monomially equivalent to a matrix-product code of a nested sequence of p^k cyclic codes with length n over R and a p^k× p^k matrix A_p^k over F_p. Using the matrix-product structures, we give an iterative construction of every (1+ω u)-constacyclic code by (1+ω u)-constacyclic codes of shorter lengths over R.

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## 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 play a very significant role in the theory of error-correcting codes.

Let be a commutative finite chain 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 .

A code of length over is a nonempty subset of , . Each element of is called a codeword and the number of codewords in is denoted by . The code is said to be linear if is a -submodule of . For any codeword , the Hamming weight of is defined by . Then the minimum Hamming distance of a linear code is equal to . If and , is called an -code over . All codes in this paper are assumed to be linear.

Let . A linear code of length over 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. It is well known that is a -constacyclic code of length over if and only if is an ideal of the residue class ring . Let be the characteristic of the residue class field of . If , is called a simple-root constacyclic code while when it is called a repeated-root constacyclic code.

For any positive integer , we denote in this paper. Let and be codes of length over . Recall that and are said to be monomially equivalent if there exists a permutation on the set and fixed elements such that

 C2={(r0cϱ(0),r1cϱ(1),…,rN−1cϱ(N−1))∣(c0,c1,…,cN−1)∈C1}

(cf. Huffman and Pless  Page 24). Especially, and are said to be permutation equivalent when (cf.  Page 20). Recall that a monomial matrix over is a square matrix with exactly one invertible entry in each row and column. Hence and are monomially equivalent if and only if there is an monomial matrix over such that in which we regard each as an

column vector over

.

From now on, let and be positive integers, a prime number, be a finite field of elements and denote

 R=Fpm[u]/⟨ue⟩=Fpm+uFpm+…+ue−1Fpm (ue=0).

It is known that is a finite chain ring with subfield , is the unique maximal ideal and is the nilpotency index of . All invertible elements of are given by

There are many research results on constacyclic codes over , see , [5–10] and  for examples. Let , and be positive integers satisfying . In this paper, we concentrate on -constacyclic codes of length over , i.e. ideals of the residue class ring . Specifically, the algebraic structures and properties of -constacyclic codes of arbitrary length over an arbitrary finite chain ring were given in , where is a unit in and generates the unique maximal ideal of .

Blackford  classified all negacyclic codes over the finite chain ring

of even length using a Discrete Fourier Transform approach. Using the concatenated structure given by  Theorem 3, we know that each negacyclic code of length

, where

is odd, is monomially equivalent to a sequence of

cyclic codes of length over .

As , negacyclic codes of even length over is a special subclass of the class of -constacyclic codes with arbitrary length over an arbitrary finite chain ring . Now, we try to give a matrix-product structure for any -constacyclic code of length over by us of the theory of finite chain rings. In this paper, we denote

 Rk:=R[v]/⟨vpk−(1+ωu)⟩.

As when , from Cao  Theorem 2.4 and Dinh et al  Section 4 we deduce the following lemma.

Lemma 1.1 Using the notations above, we have the following conclusions.

(i) is nilpotent in the ring .

(ii) is a commutative finite chain ring with maximal ideal , and is the nilpotency index of . Furthermore, .

(iii) .

(iv) All distinct ideals of are given by

 {0}=(v−1)pkeRk⊂(v−1)pke−1Rk⊂…⊂(v−1)Rk⊂(v−1)0Rk=Rk.

Moreover, the number of elements in is equal to for all .

We will construct a precise isomorphism of rings from onto , which induces a one-to-one correspondence between the set of -constacyclic codes of length over onto the set of cyclic codes of length over . By the theory of simple-root cyclic codes over finite chain rings (cf. Norton et al ), any cyclic code of length over can be determined uniquely by a tower of cyclic codes with length over the finite field

 ⟨g0(x)⟩⊆⟨g1(x)⟩⊆…⊆⟨gpke−1(x)⟩⊆Fpm[x]/⟨xn−1⟩,

where are monic divisors of in satisfying . Then we give a direct description of a monomially equivalence between a -constacyclic code of length over and a matrix-product code of a sequence of cyclic codes over determined by , .

In Section 2, we sketch the concept of matrix-product codes and structures of simple-root cyclic codes over the finite chain ring . In Section 3, we prove that any -constacyclic code of length over is monomially equivalent to a matrix-product code of a nested sequence of cyclic codes with length over . Using this matrix-product structure, we give an iterative construction of every -constacyclic code by use of -constacyclic codes of shorter lengths over in Section 4. In Section 5, we consider how to get the matrix-product structures of -constacyclic codes of length over ().

## 2 Preliminaries

In this section, we sketch the concept of matrix-product codes and structures of simple-root cyclic codes over the finite chain ring .

Let . We follow the notation in  Definition 2.1 for definition of matrix-product codes. Let be an matrix with entries in and let be codes of length over . The matrix-product code is the set of all matrix products defined by

 [c1,…,cα]⋅A = [c1,…,cα]⎡⎢ ⎢ ⎢ ⎢ ⎢⎣a11a12…a1βa21a22…a2β⋮⋮⋮⋮aα1aα2…aαβ⎤⎥ ⎥ ⎥ ⎥ ⎥⎦ = [a11c1+a21c2+…+aα1cα,a12c1+a22c2+…+aα2cα, …,a1βc1+a2βc2+…+aαβcα]

where is an column vector for . Any codeword is an matrix over and we regard it as a codeword of length by reading the entries of the matrix in column-major order. A code over is a matrix-product code if for some codes and a matrix .

In the rest of this paper, we assume that is an matrix over , i.e. for all . If the rows of are linearly independent over , is called a full-row-rank (FRR) matrix. Let be the matrix consisting of the first rows of . For , we denote by the submatrix consisting of the columns of . If every sub-matrix of is non-singular for all , is said to be non-singular by columns (NSC) (cf.  Definition 3.1).

As a natural generalization of  Theorem 1 and results in , by  Theorem 3.1 we have the following properties of matrix-product codes.

Theorem 2.1 Let be an FRR matrix over , and be a linear -code over for all . Then the matrix-product code is a linear -code over where the minimum Hamming distance satisfies

 d≥δ:=min{δidi∣i=1,…,α},

where is the minimum distance of the linear code with length over generated by the first rows of the matrix .

Moreover, when the matrix is NSC, it holds that . Furthermore, if we assume that the codes form a nested sequence , then .

Then we consider cyclic codes of length over the finite chain ring , i.e. ideals of the residue class ring . Let . By Lemma 1.1 and properties of finite chain rings, has a unique -expansion

 α=pke−1∑s=0as(v−1)s, as∈Fpm, s=0,1,…,pke−1.

In this paper, we define by

 τ(α)=a0=α (mod v−1), ∀α∈Rk.

Then is a surjective homomorphism of rings from onto . As by Lemma 1.1(ii), there is an invertible element such that , which implies . Hence for any where , we have

 τ(β)=b0=β (mod u). (1)

It is clear that can be extended to a surjective homomorphism of polynomial rings from onto by: We still use to denote this homomorphism. Then induces a surjective homomorphism of rings from onto in the natural way

 τ(n−1∑i=0αixi)=n−1∑i=0τ(αi)xi, ∀α0,α1,…,αn−1∈Rk.

Now, let be a cyclic code of length over . For any integer , , define

 (C:(v−1)s)={α(x)∈Rk[x]/⟨xn−1⟩∣(v−1)sα(x)∈C}

which is an ideal of as well. It is clear that

 C=(C:(v−1)0)⊆(C:(v−1))⊆…(C:(v−1)pke−1). (2)

Denote

 Tors(C)=τ(C:(v−1)s)={τ(α(x))∣α(x)∈(C:(v−1)s)}.

Then is an ideal of the ring , i.e. a cyclic code of length over , which is called the th torsion code of . Hence there is a unique monic divisor of in such that

 Tors(C)=⟨gs(x)⟩={b(x)gs(x)∣deg(b(x))

where is the generator polynomial of the cyclic code . Hence .

As , we have for some . Then by in , it follows that

 (v−1)sgs(x)pke−s = (v−1)s(gs(x)pke−s−(v−1)pke−sbs(x)pke−s) = (v−1)s(gs(x)−(v−1)bs(x)) ⋅⎛⎝pke−s−1∑t=0gs(x)t⋅((v−1)bs(x))pke−s−1−t⎞⎠.

This implies . As , has no repeated divisors in . This implies . Hence there exist such that in . Therefore, we have

 (v−1)sgs(x)=a(x)⋅(v−1)sgs(x)pke−s∈C, s=0,1,…,pke−1. (3)

This implies for all . Moreover, by Equation (2) we have a tower of cyclic codes over :

 Tor0(C)⊆Tor1(C)⊆…⊆Torpke−1(C)⊆Fpm[x]/⟨xn−1⟩.

This implies that in .

Now, let . Then . Hence there exists a unique polynomial satisfying such that . By Equation (3), it follows that . Hence .

As , there exists such that . This implies , and so .

By , there exists a unique polynomial satisfying such that . Then by Equation (3), it follows that . By , there exists such that and

 (v−1)2α2(x)=(v−1)α1(x)−(v−1)b1(x)g1(x)∈C.

This implies , and so .

As stated above, we have

 c(x) = b0(x)g0(x)+(v−1)α1(x) = b0(x)g0(x)+(v−1)b1(x)g1(x)+(v−1)2α2(x),

where and .

Let and assume that there exist , , and such that

 c(x)=s∑i=0(v−1)ici(x)+(v−1)s+1αs+1(x).

Then by , it follows that . This implies , and so . We denote . Then there exists such that , and hence . Therefore,

 c(x)=s+1∑i=0(v−1)ici(x)+(v−1)s+2αs+2(x).

By mathematical induction on , we conclude the following theorem.

Theorem 2.2 Using the notations above, we have the following conclusions.

(i) Let be a cyclic code of length over . Then each codeword in has a unique -adic expansion:

 c(x)=pke−1∑s=0(v−1)scs(x), where cs(x)∈Tors(C), ∀s=0,1,…,pke−1.

Hence .

(ii) is a cyclic code of length over if and only if there exists uniquely a tower of cyclic codes with length over , such that for all . If the latter conditions are satisfied, then

 C = pke−1⨁s=0(v−1)sCs = ⟨g0(x),(v−1)g1(x),…,(v−1)pkegpke−1(x)⟩Rk[x]/⟨xn−1⟩ = ⟨pke−1∑s=0(v−1)sgs(x)⟩Rk[x]/⟨xn−1⟩

where being the generator polynomial of the cyclic code for all .

Remark For a complete description of simple-root cyclic codes over arbitrary commutative finite chain rings, readers can refer to  Theorem 3.5.

When , we have satisfying or . Then from Lemma 1.1, Theorem 2.2 and Equation (1), we deduce the following corollary which will be used in the following sections.

Corollary 2.3 Using the notations above, we have the following conclusions.

(i) Let be a cyclic code of length over . Then each codeword in has a unique -adic expansion:

 c(x)=e−1∑s=0uscs(x), where cs(x)∈Tors(C)=τ(C:us), ∀s=0,1,…,e−1.

Hence .

(ii) is a cyclic code of length over if and only if there exists uniquely a tower of cyclic codes with length over , such that for all . If the latter conditions are satisfied, then and . Furthermore, we have

where being the generator polynomial of the cyclic code for all .

(iii) Let and be cyclic codes of length over with and for all . Then if and only if as ideals of the ring for all .

Proof We only need to prove (iii). If for all , it is obvious that .

Conversely, let . Then for all . From this, by and we deduce that for all .

## 3 Matrix-product structure of (1+ωu)-constacyclic codes over R

Denote . Then each integer can be uniquely expressed as

 i=j+tn, were j≡i (mod n), j∈[n), and t=i−jn∈[pk). (4)

In this paper, we adopt the following notations.

Notation 3.1 Let be the smallest positive integer such that . Since , there exists a unique integer , , such that

 n′n≡1 (mod pk+l). (5)

We write , where and satisfying . Then we define a transformation on the set by

 ϱ(j+λn)=j+n(λ−jn′′ (mod pk)), ∀(