A class of 1-generator repeated root quasi-cyclic codes

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.

Left dihedral codes over finite chain rings

1-Generator quasi-cyclic codes over [formula omitted

<abstract><p>An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.</p></abstract>

On the ℓ-DLIPs of codes over finite commutative rings

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.

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.

Let Z = F_p^m[u]/(u³) be the finite commutative chain ring, where p is a prime, m is a positive integer and Fpm is the finite field with pm elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over Z and their dual codes. We also determine the number of codewords in each repeated-root constacyclic code over Z. We also obtain Hamming distances, RT distances, RT weight distributions and ranks (i.e., cardinalities of minimal generating sets) of some repeated-root constacyclic codes over Z. Using these results, we also identify some isodual and maximum distance separable (MDS) constacyclic codes over Z with respect to the Hamming and RT metrics.

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

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.

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.

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.

For R a Galois ring and m 1, . . . , m l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ is an R[x]-submodule of $${R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}$$. Suppose m 1, . . . , m l are all coprime to the characteristic of R and let {g 1, . . . , g t } be the set of all monic basic irreducible polynomials in the factorizations of $${x^{m_i}-1}$$ (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ are identified with $${{\mathcal G}_1\times\cdots\times {\mathcal G}_t}$$, where $${{\mathcal G}_j}$$ is an R[x]/(g j )-submodule of $${(R[x]/(g_j))^{n_j}}$$, where n j is the number of i for which g j appears in the factorization of $${x^{m_i}-1}$$ into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.

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.

Let p≠3 be any prime. In this paper, we compute symbol-pair distance of all γ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm, where γ is a unit of R which is not a cube in Fpm. We give the necessary and sufficient condition for a symbol-pair γ-constacyclic code to be an MDS symbol-pair code. Using that, we provide all MDS symbol-pair γ-constacyclic codes of length 3ps over R. Some examples of the symbol-pair distance of γ-constacyclic codes of length 3ps over R are provided.