# σ-self-orthogonal constacyclic codes of length p^s over F_p^m+u F_p^m

In this paper, we study the σ-self-orthogonality of constacyclic codes of length p^s over the finite commutative chain ring F_p^m + u F_p^m, where u^2=0 and σ is a ring automorphism of F_p^m + u F_p^m. First, we obtain the structure of σ-dual code of a λ-constacyclic code of length p^s over F_p^m + u F_p^m. Then, the necessary and sufficient conditions for a λ-constacyclic code to be σ-self-orthogonal are provided. In particular, we determine the σ-self-dual constacyclic codes of length p^s over F_p^m + u F_p^m. Finally, we extend the results to constacyclic codes of length 2 p^s.

## Authors

• 12 publications
• 1 publication
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## Abstract

In this paper, we study the -self-orthogonality of constacyclic codes of length over the finite commutative chain ring , where and is a ring automorphism of . First, we obtain the structure of -dual code of a -constacyclic code of length over . Then, the necessary and sufficient conditions for a -constacyclic code to be -self-orthogonal are provided. In particular, we determine the -self-dual constacyclic codes of length over . Finally, we extend the results to constacyclic codes of length .

Keywords: constacyclic code; repeated-root code; -self-orthogonal code; -self-dual code; finite commutative chain ring.

2010 Mathematics Subject Classification.   94A55, 94B05

## 1 Introduction

The study of constacyclic codes originated in the 1960s. Berlekamp[3, 4] introduced the concept of negacyclic codes over finite fields. Constacyclic codes are the natural generalization of cyclic codes which can be technically implemented by shift registers. They have similar algebraic structure to cyclic codes so that they inherit most of the good properties of cyclic codes. The properties of constacyclic codes are easy to analyze so that they can easily be encoded and decoded. Thus, this family of codes is interesting for both theoretical and practical reasons.

Codes over finite rings have received much attention recently after it was proved that some important families of binary non-linear codes are in fact images under a Gray map of linear codes over (see, for example, [6, 17, 19]). If the characteristic of the finite ring is relatively prime to the length of a constacyclic code, we call this code a simple-root code; otherwise it is called a repeated-root code. Dinh and López-Permouth[13] obtained the structure of simple-root cyclic and negacyclic codes of length and their duals over a finite chain ring and gave necessary and sufficient conditions for the existence of a simple-root cyclic self-dual code over a finite chain ring. Since the decomposition of polynomials over finite rings is not unique, the structure of repeated-root constacyclic codes over finite rings is more complex. Since 2003, some special classes of repeated-root constacyclic codes over certain finite chain rings have been studied by many authors (see, for example,[5, 14, 15, 20, 11, 18]).

In 1997, Bachoc[2] discussed linear codes over ( or , is a prime). This work has aroused the interest of researchers in studying error correcting codes over finite chain rings of the form ( is a prime). Dinh[9] studied all constacyclic codes of length over . The algebraic structure of all constacyclic codes of length and over the finite commutative chain ring was determined in [10, 8].

Self-orthogonal codes over finite rings or finite fields are a class of important linear codes which are closely related to combinatorial designs and modular lattices. It has been found that the problem of finding quantum error-correcting codes can be transformed into the problem of finding additive codes over that are self-orthogonal with respect to a certain trace inner product [7]. This has caused a great interest in constructing classical self-orthogonal codes. Self-dual codes are a special class of self-orthogonal codes. A self-dual code has the same weight distribution as its dual code. A large number of good codes are self-dual codes. So they have been an important subject in the research of error-correcting codes. As far as we know, there have been very few results concerning self-orthogonal constacyclic codes over . Recently, all self-dual constacyclic codes of length over the finite commutative chain ring and the number of each type of self-dual constacyclic code were established in [12]. But it is not easy to obtain the the self-orthogonality from self-duality.

Let be a finite commutative Frobenius ring with an identity and be the ring automorphism group of . Let , then can be extended to a bijective map

 Rn ⟶Rn, (r0,r1,⋯,rn−1) ⟼(σ(r0),σ(r1),⋯,σ(rn−1)).

Given -tuples , their -inner product is defined as

 ⟨x,y⟩σ=n−1∑i=0xiσ(yi)=x0σ(y0)+x1σ(y1)+⋯+xn−1σ(yn−1).

When is a finite field of order , -inner product is just the usual Euclidean inner product if is the identity map of , -inner product is the Hermitian inner product if is even and maps any element of to and -inner product is the Galois inner product [16] if maps any element of to for some integer .

-inner product over finite commutative Frobenius rings generalizes the Euclidean inner product, the Hermitian inner product and Galois inner product over finite fields. Two -tuples x and y are called -orthogonal if . For a code over , its -dual code is defined as

 C⊥σ={x|⟨c,x⟩σ=0,∀c∈C}.

A code is called -self-orthogonal if , and it is called -self-dual if .

This paper focuses on the -self-orthogonality of constacyclic codes of length over the finite commutative chain ring .

The remainder of this paper is organized as follows. Preliminary concepts and some known results are given in Section . In Section , we obtain the structure of -dual codes of -constacyclic codes of length over . In Section , we provide necessary and sufficient conditions for a -constacyclic code to be -self-orthogonal using the relation between the polynomials of the generating sets of a -constacyclic codes and its -dual code. In particular, we obtain the -self-dual constacyclic codes over . The results in Section can be extended to constacyclic codes of length over .

## 2 Preliminaries

Let be a finite commutative Frobenius ring with an identity. We call a nonempty subset of a code of length over and the ring is referred to as the alphabet of . If is an -submodule of , then is said to be linear. It is easy to obtain the following proposition.

###### Proposition 2.1.

Let be a code of length over , then

(1) is a linear code over .

(2) . Moreover, if is a linear code, then .

###### Proof.

(1) For any and we have

 ⟨x,r1y1+r2y2⟩σ=n−1∑i=0xiσ(r1y1i+r2y2i)=σ(r1)n−1∑i=0xiσ(y1i)+σ(r2)n−1∑i=0xiσ(y2i)=σ(r1)⟨x,y1⟩σ+σ(r2)⟨x,y2⟩σ=0.

Thus, , which means that is a linear code over .

(2) For any , we have if and only if if and only if if and only if , implying that , where is the usual Euclidean inner product.

Since can be extended to a bijective map from to , If is linear, then . Hence, . ∎

For a unit of , the -constacyclic (-twisted) shift on is the shift

 τλ(x0,x1,⋯,xn−1)=(λxn−1,x0,x1,⋯,xn−2).

A linear code is said to be -constacyclic if . The -constacyclic codes are the cyclic codes and the -constacyclic codes are just the negacyclic codes.

Let be a polynomial over and let denote the degree of . Under the standard -module isomorphism

 Rn ⟶R[x]/⟨xn−λ⟩, (c0,c1,⋯,cn−1) ⟼c0+c1x+⋯+cn−1xn−1+⟨xn−λ⟩,

each codeword can be identified with its polynomial representation

 c(x)=c0+c1x+⋯+cn−1xn−1∈R[x],degc(x)⩽n−1,

and each -constacyclic code of length over can also be viewed as an ideal of the quotient ring In the light of this, the study of -constacyclic codes of length over is equivalent to the study of ideals of the quotient ring It is easy to prove the following proposition by Proposition 2.1.

###### Proposition 2.2.

The -dual code of a -constacyclic code is a -constacyclic code.

Let , where . Then the polynomial is called the reciprocal polynomial of . In fact, can also be expressed as . We can see that if is an ideal of , then is an ideal of .

Let be an ideal of . The annihilator of denoted by is defined as

 A(I)={g(x)∈R[x]/⟨xn−λ⟩|f(x)g(x)=0,∀f(x)∈I}.

Then is also an ideal of . It is clear that if is a -constacyclic code of length over , then is and is .

Throughout this paper, let

be an odd prime and

be a positive integer. denotes the finite field of order , where is a positive integer. denotes the multiplicative cyclic group of non-zero elements of . Let , where . Then is a finite commutative chain ring with the unique maximal ideal , whose ideals are , and . Each element of can be expressed as , where . Then element is a unit of if and only if . If , then is a square of if and only if is a square of . The automorphism group of is given as follows.

###### Proposition 2.3.

([1]) For and , let

 Θθ,ε:  R ⟶R, a+ub ⟼θ(a)+εθ(b).

Then .

In the rest of this paper, let . Then , i.e., for some and .

For a code of length over , its torsion and residue codes are defined as follows.

 Tor(C)={b∈Fnpm|ub∈C},Res(C)={a∈Fnpm|∃ b% ∈Fnpm such that a+ub∈C}.

Then both of them are codes of length over . The reduction modulo from to is defined as

 ϕ:  C ⟶Res(C), a+ub ⟼a.

Clearly, is well defined and surjective, with , . Therefore, .

## 3 The structure of σ-dual codes of λ-constacyclic codes

The algebraic structure of all -constacyclic codes of length over was obtained in [10]. The situation of is divided into two cases separately: (a) , where , are nonzero elements of , (b) , where is a nonzero element of .

First, we consider the case that , where , are nonzero elements of .

Let , then the -constacyclic codes of length over are ideals of the ring . By the division algorithm, there exist nonnegative integers , such that , and . Let . Then . We have the following conclusions:

###### Lemma 3.1 ([10]).

In , . In particular, is nilpotent in with nilpotency index . is a chain ring with ideals that are precisely

 Rα,β=⟨1⟩⫌⟨α0x−1⟩⫌⋯⫌⟨(α0x−1)2ps−1⟩⫌⟨(α0x−1)2ps⟩=⟨0⟩.

-constacyclic codes of length over are the ideals , of the chain ring . Each code contains codewords.

###### Theorem 3.1.

Let be an -constacyclic code of length over and for some , then its -dual code is the -constacyclic code

 C⊥σ=⟨(θ(α−10)x−1)2ps−i⟩⊆Rθ(α−1),−εθ(βα−2),

which contains codewords.

###### Proof.

It follows immediately from Proposition 2.1 and Theorem 4.3 of [10]. ∎

In the following, we discuss the case that , where is a nonzero element of .

Let , then the -constacyclic codes of length over are ideals of the ring . By the division algorithm, there exist nonnegative integers , such that , and . Let . Then .

In [10] the authors studied -constacyclic codes by constructing a one-to-one correspondence between cyclic and -constacyclic codes as follows.

###### Proposition 3.1 ([10]).

The map given by is a ring isomorphism. In particular, for with , then is an ideal of if and only if is an ideal of . Equivalently, is a cyclic code of length over if and only if is a -constacyclic code of length over .

Now, using the isomorphism , the results about cyclic code of length over in [10] can be applied to corresponding -constacyclic codes of length over .

###### Lemma 3.2 ([10]).

is nilpotent in with nilpotency index . The ring is a local ring with the maximal ideal , but it is not a chain ring.

Then all the -constacyclic codes of length over , i.e., ideals of the local ring

, are classified into four types, as follows.

###### Theorem 3.2 ([10]).

-constacyclic codes of length over , i.e., ideals of the local ring , are:

• Type 1 (trivial ideals): , .

• Type 2 (principal ideals with nonmonic polynomial generators): , where .

• Type 3 (principal ideals with monic polynomial generators):

 ⟨(γ0x−1)i+u(γ0x−1)th(x)⟩,

where , and either is or is a unit where it can be represented as , and .

• Type 4 (nonprincipal ideals): , where , , where is the smallest integer such that , with as in Type 3, and .

For -constacyclic codes of Type 4 in Theorem 3.2, the number plays a very important role. We now determine for each code .

###### Proposition 3.2 ([10]).

Let be the smallest integer such that

 u(γ0x−1)T∈⟨(γ0x−1)i+u(γ0x−1)th(x)⟩,

then

 T={i,if h(x)=0;min{i,ps−i+t},if h(x)≠0.

We now compute the size of each -constacyclic code . By the definitions of and and , we have the following result.

###### Theorem 3.3 ([10]).

is -constacyclic codes of length over , as classified in Theorem 3.2. Then the number of codewords of is determined as follows.

• If , then .

• If , then .

• If , where , then .

• If , where , then .

• If , where , , and is a unit, then

 nC=⎧⎨⎩p2m(ps−i),if 1≤i≤ps+t2;pm(ps−t),if ps+t2
• If , where , , is or a unit, and

 κ

then .

###### Theorem 3.4.

Let be a -constacyclic code of length over .

• If , then . If , then .

• If , where , then .

• If , where , , is is a unit which can be represented as , and .

(1) If , then .

(2) If is a unit, and , then where

 a(x)=(θ(γ−10)x−1)ps−i−u(θ(γ−10)x−1)ps+t−2ii−t−1∑j=0εθ(hj)(−1)j+t−i(θ(γ−10)x−1)jxi−j−t.

(3) If is a unit, and , then where

 b(x)=(θ(γ−10)x−1)i−t−ups−i−1∑j=0εθ(hj)(−1)j+t−i(θ(γ−10)x−1)jxi−j−t.
• If where , , is the smallest integer such that , is defined as Type 3 in Theorem 3.2, and .

(1) If , then .

(2) If is a unit, then where

 d(x)=(θ(γ−10)x−1)ps−ω−u(θ(γ−10)x−1)ps−i−ω+tω−t−1∑j=0εθ(hj)(−1)j+t−i(θ(γ−10)x−1)jxi−j−t.
###### Proof.

It follows immediately from Proposition 2.1 and Theorem 4.7, Theorem 4.8, Theorem 4.9 of [12]. ∎

## 4 σ-self-orthogonal λ-constacyclic codes of length ps

In this section, we study the -self-orthogonality of the -constacyclic codes of length over . We provide necessary and sufficient conditions for a -constacyclic code to be -self-orthogonal using the relation between the polynomials of the generating sets of a -constacyclic codes and its -dual code. In particular, we get -self-duality of the constacyclic codes over from their -self-orthogonality.

Consider the code of length over . Clearly, for any unit of , is -constacyclic code of length over . It is also the ideal of generated by . can also denote this ideal.

### 4.1 σ-self-orthogonal (α+uβ)-constacyclic codes

, where , are nonzero elements of . .

###### Theorem 4.1.

Let be an ()-constacyclic code of length over and for some , then is -self-orthogonal if and only if .

###### Proof.

By Theorem 3.1, .

Necessity. If is -self-orthogonal, then , which yields It follows from Theorem 3.1 that . That means that , i.e., . Since , we have .

Sufficiency. From we have . Therefore,

 C=⟨(α0x−1)i⟩=⟨(α0x−1)ps(α0x−1)i−ps⟩=⟨u(α0x−1)i−ps⟩⊆⟨u⟩=⟨(θ(α−10)x−1)ps⟩⊆⟨(θ(α−10)x−1)2ps−i⟩=C⊥σ,

which implies that is -self-orthogonal. ∎

###### Theorem 4.2.

is the unique -self-dual ()-constacyclic code of length over .

###### Proof.

Since , It follows from Theorem 4.1 that is -self-orthogonal, i.e., . By Theorem 3.1, , , which means that . Hence, , i.e., is -self-dual.

Next, we will prove the uniqueness. Assume that is -self-dual, then . That follows that . In view of Theorem 3.1, , . This leads to , i.e., , implying that . ∎

### 4.2 σ-self-orthogonal γ-constacyclic codes

, where is a nonzero element of .