## 1 Introduction

The catalyst for the study of codes over rings was the discovery of the connection between the Kerdock and Preparata codes, which are non-linear binary codes, and linear codes over (see [3] and [4]). Soon after this discovery, codes over many different rings were studied. This led to many new discoveries and concreted the study of codes over rings as an important part of the coding theory discipline. Since is a chain ring, it was natural to expand the theory to focus on alphabets that are finite commutative chain rings and other special rings (See [1], [2], [5], [7–12], [14], [16–22], for examples).

In 1999, Wood in [23] showed that for certain reasons finite Frobenius rings are the most general class of rings that should be used for alphabets of codes. Then self-dual codes over commutative Frobenius rings were investigated by Dougherty et al. [13]. Especially, in 2014, codes over an extension ring of were studied in [24] and [25], here the ring was described as () which is a local non-principal ring.

In this paper, all rings are associative and commutative. Let be an arbitrary finite ring with identity , the multiplicative group of units of and . 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 ) in this paper.

For any positive integer , let which is an -module with componentwise addition and scalar multiplication by elements of . Then an -submodule of is called a linear code of length over

. For any vectors

. The usual Euclidian inner product of and is defined by . Let be a linear code over of length . The dual code of is defined by , and is said to be self-dual if .A linear code over of length is said to be negacyclic if

We will use the natural connection of negacyclic codes to polynomial rings, where , is viewed as and the negacyclic code is an ideal in the polynomial residue ring .

In this paper, let be an odd positive integer and denote

in which the operations are defined by:

and .

for any with . Then is a local Frobenius non-principal ideal ring of elements.

Linear codes over were studied in [15]. In the paper, a duality preserving Gray map was given and used to present MacWilliams identities and self-dual codes. Some extremal Type II -codes were provided as images of codes over this ring. -codes that are images of linear codes over were characterised and some well-known families of -codes were proved to be linear over . As in [15] Section 3, we define a map by

and let be such that , for all Let denote the Lee weight on defined by:

We extend on the ring in a natural way that

With this distance and Gray map definition, the following conclusions have been verified by Martínez-Moro et al. [15].

Lemma 1.1 ([15] Theorem 3.1) Let be a linear code over of length and minimum Lee distance . Then is a linear code over of length , and is of minimum Lee distance .

Lemma 1.2 ([15] Proposition 3.3) Let be a linear code over of length . Then . In particular, if is self-dual, then is an self-dual code over of length and has the same Lee weight distribution.

Moreover, we have the following properties for Negacyclic codes .

Proposition 1.3 Let be a negacyclic code of length . Then is a -quasi-twisted code over of length .

Proof. Let , where , , for all . Then . As is negacyclic, we have . This implies . So is a -quasi-twisted -code of length .

Since is odd, the map defined by

is an isomorphism of rings preserving Lee distance. Hence is a negacyclic code over of if and only if there is a unique cyclic code over of length such that . Moreover, and has the same Lee weight distribution. A complete classification for cyclic codes over of odd length and self-dual codes among them had been studied in [5]. In the paper, some good self-dual codes over of length and extremal binary self-dual codes with parameters were obtained from self-dual cyclic codes over of length . In this paper, we study negacyclic codes over of length .

The present paper is organized as follows. In Section 2, we sketch the basic theory of finite rings and linear codes over finite rings needed in this paper. In Section 3, we decompose the ring into a direct product of finite chain rings of length . In Section 4, we give a canonical form decomposition for any negacyclic code over of length and present all distinct codes by their generator sets. Using this decomposition, we give the number of codewords for each of these codes and an enumeration for all these codes. In Section 5, we present the dual code and its self-duality for each negacyclic code over of length . In Section 6, we focus our attention on negacyclic code over of length , where is a prime and is a Mersenne prime. Especially, we present explicitly all ngeacyclic code over of length and self-dual codes among them. Finally, we obtain new and good self-dual -quasi-twisted codes over of length .

## 2 Preliminaries

In this section, we sketch the basic theory of finite chain rings and linear codes over finite chain rings needed in this paper.

Lemma 2.1 ([10] Proposition 2.1) Let be a finite ring with identity. Then the following conditions are equivalent:

(i) is a local ring and the maximal ideal of is principal, i.e. for some ;

(ii) is a local principal ideal ring;

(iii) is a chain ring with all ideals given by: , , where is the nilpotency of .

Lemma 2.2 ([10] Proposition 2.2) Let be a finite chain ring, with maximal ideal , and let be the nilpotency of . Then

(i) For some prime and positive integer , where , , and the characteristic of and are powers of ;

(ii) For , .

Lemma 2.3 ([16] Lemma 2.4)Using the notations in Lemma 2.2, let be a system of representatives for the equivalence classes of under congruence modulo . (Equivalently, we can define to be a maximal subset of with the property that for all , .) Then

(i) Every element of has a unique -expansion: , .

(ii) and for .

From now on, let be an arbitrary finite chain ring with , be a fixed generator of the maximal ideal of with nilpotency index , and . In this case, is called a finite chain ring of length . Using the notations of Lemma 2.3, every element has a unique -adic expansion:

Hence . If , the -degree of is defined as the least index for which and written for . If we write . It is clear that if and only if , i.e. . Hence . Moreover, we have and with , .

Let be a positive integer and that is a free -module under componentwise addition and scalar multiplication with elements from . Then -submodules of are linear codes over of length . Let be a linear code over of length . By [16] Definition 3.1, a matrix is called a generator matrix for if the rows of span and none of them can be written as a -linear combination of the other rows of . Furthermore, a generator matrix is said to be in standard form if there is a suitable permutation matrix of size such that

(1) |

where the columns are grouped into blocks of sizes with and . Of course, if , the matrices and () are suppressed in . From [16] Proposition 3.2 and Theorem 3.5, we deduce the following.

Lemma 2.4 Let be a nonzero linear code of length over . Then has a generator matrix in standard form as in . In this case, the number of codewords in is equal to .

All distinct nontrivial linear codes of length over has been listed (cf. Cao [6] Lemma 2.2 and Example 2.5). In particular, we have

Theorem 2.5 Using the notations above, let . Then every linear code over of length satisfying the following condition

(2) |

has one and only one of the following matrices as its generator matrix in standard form:

(I) , where .

(II) ; where ; where .

(IV) ; where and .

(V) where , and where .

Therefore, the number of linear codes over of length satisfying Condition is equal to .

Proof. See Appendix.

## 3 A direct sum decomposition of the ring

From now on, let be an odd positive integer. In this section, we decompose the ring into a direct product of finite chain rings of length . This direct sum decomposition will be needed in the following sections.

It is known that any element of is unique expressed as where in which we regard as a subset of . Denote . Then () is a ring isomorphism from onto , and can be extended to a ring isomorphism from onto by: for all

A monic polynomial is said to be basic irreducible if is an irreducible polynomial in . Then we have the following conclusions for monic basic irreducible polynomials in .

Lemma 3.1 Let be a monic basic irreducible polynomial in of degree and denote . Then

(i) ([22] Theorem 6.1]) is a Galois ring of characteristic and cardinality and , where satisfying in .

Denote and . Then which is a finite field of cardinality , and that can be extended to a ring isomorphism from onto by , for all where .

(ii) ([22] Proposition 6.14 or [7] Lemma 2.3(ii)) .

Theorem 3.2 Let be a monic basic irreducible polynomial in of degree , denote in which the arithmetic is done modulo , and set in which we regard as a subset of . Then

(i) There is an invertible element such that . Hence as ideals of .

(ii) is a finite chain ring with maximal ideal generated by , the nilpotency index of is equal to and is a finite field of cardinality .

(iii) Every element of has a unique -adic expansion given by:

Moreover, we have as sets and .

Proof. (i) By Lemma 3.1(ii), we have

By Lemma 3.1(i), we know that . From this, by Lemma 3.1(ii) and we deduce that

where , since , as a polynomial in we have that

This implies for all , , since are distinct root of in the finite field by Lemma 3.1(i). From this and by , we deduce that in . Therefore, and are coprime in . Hence there exist such that This implies that is an invertible element of the residue class ring and (mod ). Then by in it follows that in . Hence as ideals of .

(ii) Let be the ideal of generated by and . Then

up to natural ring isomorphisms, where is a finite field of elements by Lemma 3.1(i). Hence is a maximal ideal of .

By , we see that both and are nilpotent elements of . From this one can verify easily that every element of is invertible. This implies that is a local ring with as its unique maximal ideal. Furthermore, by in (i) we conclude that . Hence .

As stated above, by Lemma 2.1 we see that is a finite chain ring. Let be the nilpotency index of . By Lemma 2.2(i) it follows that . On the other hand, by it follows that . Therefore, .

(iii) By , we see that as sets. Hence is a system of representatives for the equivalence classes of under congruence modulo . Then the conclusion follows from Lemma 2.3(i) immediately.

In the rest of this paper, let

(3) |

where are pairwise coprime monic basic irreducible polynomials in . We assume and denote

for each integer , . Then by Theorem 3.2, we know that

There is an invertible element such that

(4) |

where as a polynomial in (mod , mod ). Hence as ideals of .

is a finite chain ring with maximal ideal , the nilpotency index of is equal to and is a finite field of cardinality .

Every element of has a unique -adic expansion: , where for all . Moreover, we have as sets, and .

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

(5) |

Substituting for in (3) and (5), we obtain

and in the ring respectively. In the rest of this paper, we set

(6) |

Then from classical ring theory, we deduce the following conclusions.

Theorem 3.3 Denote . We have the following:

(i) , and in the ring , for all .

(ii) , where and its multiplicative identity is . Moreover, this decomposition is a direct sum of rings in that for all integers and , .

(iii) For each , define a mapping . Then is a ring isomorphism from onto . Hence .

(iv) Define , i.e.

for all , . Then is a ring isomorphism from onto .

## 4 Structure of negacyclic codes over of length

In this section, we list all distinct negacyclic codes of length over the ring (), i.e. all distinct ideals of the ring . Using the notation of Theorem 3.3, we denote

in which the arithmetic is done modulo , and set

in which the operations are defined by: for any ,

,

.

Then is a finite commutative ring containing as its subring.

Let . Then and can be uniquely expressed as and respectively, where for all . Now, we define a map by

Then one can easily verify the following conclusion.

Lemma 4.1 The map defined above is an isomorphism of rings from

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