Some constacyclic codes over finite chain rings

Let p ≠ 3 be any prime. The structures of all λ-constacyclic codes of length 3p^s over the finite commutative chain ring Fp^m + uFp^m (u² = 0) are established in the term of their generator polynomials. As an application, Hamming and homogeneous distance of a class of such codes and RT distances of all are given. Among such λ-constacyclic codes, the unique maximum-distance-separable (briefly, MDS) code with respect to the RT distance is obtained. Moreover, when λ is not a cube in Fpm, the necessary and sufficient condition for the λ-constacyclic code of length 3p^s over Fp^m + uFp^m (u² = 0) be an MDS constacyclic code with respect to Hamming distance is provided.

Let [Formula: see text] be a Type 1 unit in [Formula: see text] where [Formula: see text] is an odd prime, [Formula: see text] is a positive integer and [Formula: see text] In this paper, we give the complete structure of all Type 1 [Formula: see text]-constacyclic codes and their duals of length [Formula: see text] over the finite commutative chain ring [Formula: see text] in terms of their generator polynomials. Using this structure, we determine the Hamming distance and the Rosenbloom–Tsfasman (RT) distance of all Type 1 [Formula: see text]-constacyclic codes. For [Formula: see text] and a unit [Formula: see text] we determine the [Formula: see text] -symbol distances of all [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] As illustrations, we provide several [Formula: see text]-constacyclic codes with new parameters with respect to Hamming, RT and [Formula: see text]-symbol metrics. MDS codes are widely recognized for their optimal error-correction capability, and MDS [Formula: see text]-symbol codes are generalization of MDS codes. We found some MDS [Formula: see text]-symbol constacyclic codes of length [Formula: see text] over [Formula: see text] Additionally, for [Formula: see text] we provide a decoding algorithm for Type 1 constacyclic codes of length [Formula: see text] over [Formula: see text] with respect to the Hamming, RT and [Formula: see text]-symbol metrics.

The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacyclic codes over the finite chain ring R. Through the paper, it is assumed that the length of codes n can not be divided by the characteristic of R.

Left dihedral codes over finite chain rings

On the ℓ-DLIPs of codes over finite commutative rings

In this paper, we study constacyclic codes over finite principal ideal rings. An isomorphism between constacyclic codes and cyclic codes over finite principal ideal rings is given. Further, an open question is partially answered by giving necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings. As an example of codes over a finite principal ideal ring, we study constacyclic codes over R+vR where v2=v and R is a finite chain ring.

We introduce (1-γ)-cyclic code and cyclic codes over the finite chain ring R. We prove that the Gray image of a linear (1-γ)-cyclic code over R of length n is a distance invariant quasi-cyclic code over Fpk. We also prove that if (n,p) = 1, then every code over Fpk which is the Gray image of a cyclic code over R of length n is equivalent to a quasi-cyclic code.

We use group algebra methods to study cyclic codes over finite chain rings and under some restrictive hypotheses, described in section 2, for codes of length $ 2p^n $, $ p $ a prime, we are able to compute the minimum weights of all possible cyclic codes of that length.

The main objective of coding theory is to find codes with efficient encoding and decoding techniques and the greatest possible value of distance for a given code length, code size, and code alphabet cardinality. Various distances like Hamming distances, symbol-pair distances, etc., have been established and investigated in coding theory to analyse a code’s error-detecting and error-correcting capabilities with respect to various communication channels. A maximum distance separable code (MDS code) is a code that meets the Singleton bound. The MDS codes have the greatest error correcting capability. When the length and size of the codes are fixed, an MDS code has the largest distance. As a result, nowadays, researching and discovering MDS codes with respect to various distances is a hot topic in coding theory. In the theory of error-correcting codes, constacyclic codes over finite fields play a crucial role. Constacyclic codes, moreover, have many real-world applications. Shift registers can effectively encode and decode these codes because they have rich algebraic structures. They also offer excellent error-correction capabilities. All of this clarifies their chosen engineering role. In this monograph, we determine the multi-symbol distances of various classes of repeated-root constacyclic codes over some classes of finite commutative chain rings and investigate various classes of non-trivial MDS codes using these multi-symbol distance distributions.

In this article, the notion of γ-dual codes over finite chain rings is introduced as an extension of dual codes over finite chain rings. Various characteristics and properties of γ-dual codes over finite chain rings are explored. We provide both necessary and sufficient conditions for the existence of γ-self-dual codes over finite chain rings. Additionally, we investigate the γ-dual of skew ϕ-α-constacyclic codes over finite chain rings.

Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n n\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n n\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Cyclic [formula omitted]-additive codes

In this paper, codes over the direct product of two finite commutative chain rings are studied. The standard form of the parity-check matrix is determined. The structure of self-dual codes is described. A distance preserving Gray map from the direct product of chain rings to a finite field is defined. Two upper bounds on minimum distance are obtained.

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including $${\mathbb{Z}_{p^e}}$$ ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 e , l) of length n = 2 l for any a ? 1 and l ? 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.

Let m, e be positive integers, p a prime number, $$\mathbb {F}_{p^m}$$ be a finite field of $$p^m$$ elements and $$R=\mathbb {F}_{p^m}[u]/\langle u^e\rangle $$ which is a finite chain ring. For any $$\omega \in R^\times $$ and positive integers k, n satisfying $$\mathrm{gcd}(p,n)=1$$ , we prove that any $$(1+\omega u)$$ -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 $$p^k\times p^k$$ matrix $$A_{p^k}$$ over $$\mathbb {F}_p$$ . Using the matrix-product structures, we give an iterative construction of every $$(1+\omega u)$$ -constacyclic code by $$(1+\omega u)$$ -constacyclic codes of shorter lengths over R.