## 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. Since 1999, special classes of constacyclic codes over certain classes of finite commutative chain rings have been studied by numerous authors (see [1], [2], [4], [5], [7–9], [12–15], [17–21], for example). It is an important way and an interesting topic to construct optimal codes (over finite fields or finite rings) from special linear codes over some appropriate rings.

Let be a finite commutative ring with identity , the multiplicative group of units of and . We denote by or 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 over of length

. For any vectors

. The usual Euclidian inner product of and is defined by . Let be a linear code over of length . The Euclidian dual code of is defined by , and is said to be self-dual if .Let . Then a linear code over of length is called a -constacyclic code if for all . Particularly, is a negacyclic code if , and is 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 and that the dual code of a -constacyclic code of length over is a -constacyclic code of length over , i.e. is an ideal of (cf. [14] Propositions 2.2 and 2.3).

In 1999, Wood in [24] 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. [16]. Especially, in 2014, codes over an extension ring of were studied by Yildiz et al. [25] and [26], here the ring was described as () which is a local non-principal ring. Then a complete classification and an explicit representation for cyclic codes of odd length over () were provided by Cao et al. [6] for any integer .

Recently, Shi et al. in [22] studied -constacyclic codes over the ring () of odd length , and properties of these codes and their images were investigated in the paper. As a further development, a complete classification for simple-root cyclic codes over non-principal rings was presented in [10] for any prime number and integer , and negacyclic codes over the local ring of oddly even length and their Gray images were studied in [11].

In 2017, Bandi et al. [3] studied negacyclic codes of length over (). Some results in [3] are the following:

() Described the ideal structure of (distinct negacyclic codes of length over ) by Theorem 11 in Pages 248–249 of [3], and given a mass formula for the number of negacylic codes of length over by Theorem 12 in Pages 249 of [3]. The proof for this mass formula takes pages: Pages 250–260.

() Described roughly the ideal structure of , where

is an odd integer, by Theorems 18–20, discrete Fourier transform and Theorem 21 in Pages 268–270 of [3]. Moreover, by Theorem 22 of Page 270 in [3] the number of distinct negacyclic codes of length

over is given by , wheredenotes a complete set of representatives of the -cyclotomic cosets modulo (see Page 270 in [3]).

For each , is the size of the -cyclotomic cosets modulo containing , is a Galois ring extension of with degree (see Page 267 in [3]) and is the number of distinct ideals in the ring (see Pages 267 and 270 in [3]).

There are three problems in ref. [3]:

The mass formula for the number of negacylic codes of length over given by Theorem 12 in [3] is wrong (see Remark 4.4 in this paper);

The number of distinct ideals in the ring had not been determined for any in [3].

Hence an explicit formula for the number of all distinct negacyclic codes over of arbitrary even length has not been obtained.

Although negacyclic codes over of arbitrary even length were studied in Section 4 of [3], but the expression for each code given by Theorems 18–20, discrete Fourier transform and Theorem 21 in Pages 268–270 of [3] is a little of complicated and not clear enough to list precisely all distinct negacyclic codes over of specific even length.

Motivated by those, we adopt a new idea and use some new methods to study negacyclic codes over of arbitrary even length.

In this paper, let in which the arithmetic is done modulo and denote in which the arithmetic is done modulo . We will regard as a subset of in this paper. But is not a subring of . Let . Then has a unique -adic expansion: , . It is well known that if and only if . Denote . Then () is a ring homomorphism from onto , and this homomorphism can be extended to a ring homomorphism from onto by:

A monic polynomial of positive degree is said to be basic irreducible if is an irreducible polynomial in (cf. 13.4 in [23]).

For any monic polynomial of degree , let in which the arithmetic is done modulo and in which the arithmetic is done modulo . In this paper, we still use to denote the homomorphism of rings from onto defined by:

In this paper, let be any positive integer and be an odd positive integer. We assume

(1) |

where are pairwise coprime monic basic irreducible polynomials in and

Then , , , are pairwise coprime irreducible polynomials in and for all . We will adopt the following notation.

in which the operations are defined by

for any with . Then a local non-principal ideal ring.

in which the arithmetic is done modulo .

Let in the following.

in which the arithmetic is done modulo , where .

with .

The present paper is organized as follows. In Section 2, we prove that each is a finite chain ring, , and establish an explicit isomorphism of rings from the direct product ring onto . In Section 3, we construct a precise isomorphism of rings from the direct product ring onto first. Then we present all distinct ideals of each ring explicitly. Hence we give an explicit representation and enumeration for negacyclic codes of length over . In Section 4, we list all distinct negacyclic codes of length over by their explicit expressions, obtain a precise formula to count the number of these codes and correct a mistake for the mass formula to the number of negacyclic codes of length over obtained in [3, Cryptogr. Commun. (2017) 9: 241–272].

## 2 Structure of the ring

In this section, we consider how to decompose the ring into a direct sum of finite chain rings first.

Lemma 2.1 ([13] Proposition 2.1) Let be a finite associative and commutative 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 ideals , , where is the nilpotency index of .

Lemma 2.2 ([18] Proposition 2.2) Let be a finite commutative chain ring, with maximal ideal , and let be the nilpotency index of . Then

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

(ii) For , .

Lemma 2.3 ([18] 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 -adic expansion: , , .

(ii) and for .

Let . From now on, we adopt the following notation.

in which the arithmetic is done modulo .

in which the arithmetic is done modulo .

Lemma 2.4 (i) (cf. [23] Theorem 14.1]) is a Galois ring of characteristic and cardinality , in symbol as . Moreover, we have , where satisfying , i.e. .

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

(ii) (cf. [5] Lemma 2.3(ii)) .

Now, we determine the algebraic structure of each ring , .

Lemma 2.5 Using the notation in Section 1, let . Then

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

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

(iii) Each element has a unique -adic expansion:

where for all .

(iv) All distinct ideals of are given by: , . Moreover, we have .

(v) Let . Then in which and .

(vi) Let . Then where we set for convenience, and .

Proof (i) By Lemma 2.4(ii), and in for all , it follows that

where satisfying (mod ). Then we have (mod ). This implies

Here, we regard as a subset of but is not a subring of .

Now, let . Then as a polynomial in , we see that

which satisfies

Since are all distinct roots in of the irreducible polynomial and , we have for all . From these, we deduce that . This implies as polynomials in . Then by , we see that and are coprime in . This implies that for some , i.e. (mod ). Hence is an invertible element in the ring . Therefore, we have as ideals of .

(ii) Let be the ideal of generated by and . Then the residue class ring of modulo is given by:

where is a finite field of elements. Hence is a maximal ideal of . Moreover, by (i) we have . This implies which is a principal ideal of generated by .

As in , we see that every element of is nilpotent. Hence each element in must be an invertible element, and so is the unique maximal ideal of of . Therefore, is a finite chain ring with the unique maximal ideal by Lemma 2.1.

Let be the nilpotency index of in . Then by Lemma 2.2. From this, by and we deduce that .

(iii) Using the notation of Section 1, we know that . As ia a monic basic irreducible polynomial in , it follows that for all satisfying . Moreover, we have . Hence is a system of representatives for the equivalence classes of under congruence modulo . Then the conclusion follows from Lemma 2.3 immediately.

(iv)–(vi) follow from properties of finite chain rings (cf. [18]). Here, we omit the proofs.

Next, we consider how to decompose the ring into a direct sum of finite chain rings.

Let and denote . As , we see that and are coprime in (cf. [23] Lemma 13.5). Hence there are polynomials such that

(2) |

In this paper, we define by:

(mod ).

Substituting for in Equations (1) of Section 1 and (2) above, we obtain

i.e. , and

Then from the definition of and the Chinese Remainder Theorem for commutative rings with identity, we deduce the following conclusion.

Theorem 2.6 Using the notation above, we have the following conclusions:

(i) , and in for all .

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

(iii) For each and , define a map by

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