Notes on linear codes over finite commutative chain rings

The properties of the generator matrix are given for linear codes over finite commutative chain rings,and the so-called almost-MDS (AMDS) codes are studied.     

Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

On self-dual cyclic codes over finite chain rings

In this paper, we give necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite chain rings. We prove that there are no free cyclic self-dual codes over finite chain rings with odd characteristic. It is also proven that a self-dual code over a finite chain ring cannot be the lift of a binary cyclic self-dual code. The number of cyclic self-dual codes over chain rings is also investigated as an extension of the number of cyclic self-dual codes over finite fields given recently by Jia et al.     

2n Bordered constructions of self-dual codes from group rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes of length 64 and 68, constructing 30 new extremal self-dual codes of length 68.     

形式幂级数环上的自对偶码

形式幂级数环R_∞=F[[γ]]={sum from l=0 to a_lγ~l|a_l∈F}与有限链环R_i={a_0+a_1γ+…+a_(i-1)γ~(i-1)|a_i∈F}的码的投影与提升有密切关系.利用形式幂级数环R_∞上码C在有限链环R_i的投影码的自正交性与自对偶性来研究码C的自正交性与自对偶性,得到了两个有意义的结果.     

有限链环上循环码与负循环码的生成多项式

从另一种角度研究了有限链环上循环码.给出了这种环上循环码的构造由这种构造得到了有限链环上的循环码的生成多项式.借助有限链环上循环码与负循环码的同构,也得到了这种环上循环码的生成元.     

Commutative weakly nil-neat group rings

Abstract

Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results, namely those established by McGovern et al. in (J. Algebra Appl., 2015), by Danchev-McGovern in (J. Algebra, 2015) and by the present authors in (J. Math., Tokushima Univ., 2019), related to commutative nil-clean, weakly nil-clean and nil-neat group rings, respectively.     

CYCLIC CODES OVER FORMAL POWER SERIES RINGS

In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.     

Quasi-cyclic codes as cyclic codes over a family of local rings

We give an algebraic structure for a large family of binary quasi-cyclic codes. We construct a family of commutative rings and a canonical Gray map such that cyclic codes over this family of rings produce quasi-cyclic codes of arbitrary index in the Hamming space via the Gray map. We use the Gray map to produce optimal linear codes that are quasi-cyclic.     

On LCD,self dual and isodual cyclic codes over finite chain rings

In this paper, LCD cyclic, self dual and isodual codes over finite chain rings are investigated. It was proven recently that a non-free LCD cyclic code does not exist over finite chain rings. Based on algebraic number theory, we introduce necessary and sufficient conditions for which all free cyclic codes over a finite chain ring are LCD. We have also obtained conditions on the existence of non trivial self dual cyclic codes of any length when the nilpotency index of the maximal ideal of a finite chain ring is even. Further, several constructions of isodual codes are given based on the factorization of the polynomial $x^n-1$ over a finite chain ring.     

Determinants of matrices over commutative finite principal ideal rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring R of a fixed determinant a is determined for all $a\in R$ and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.     

有限偶链环上类型Ⅱ码

从任意有限环上类型Ⅱ码的概念出发,借助两类有限链环为偶环的特性,研究了其上码为类型Ⅱ码的条件,得到了两个结果.     

Commutative Local Rings Whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in Behboodi et al. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules (J. Algebra 345:257–265, 2011) the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.     

Quasideterminant Characterization of MDS Group Codes over Abelian Groups

A group code defined over a group G is a subset of Gⁿ which forms a group under componentwise group operation. The well known matrix characterization of MDS (Maximum Distance Separable) linear codes over finite fields is generalized to MDS group codes over abelian groups, using the notion of quasideterminants defined for matrices over non-commutative rings.     

Cardinal rank metric codes over Galois rings

In 1985, Gabidulin introduced the rank metric in coding theory over finite fields, and used this kind of codes in a McEliece cryptosystem, six years later. In this paper, we consider rank metric codes over Galois rings. We propose a suitable metric for codes over such rings, and show its main properties. With this metric, we define Gabidulin codes over Galois rings, propose an efficient decoding algorithm for them, and hint their cryptographic application.     

A-Codes from Rational Functions over Galois Rings

In this paper, we describe authentication codes via (generalized) Gray images of suitable codes over Galois rings. Exponential sums over these rings help determine—or bound—the parameters of such codes.     

Homogeneous metric and matrix product codes over finite commutative principal ideal rings

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.     

Invariants of finite Hopf algebras

This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Topics considered include integrality over the invariant rings, properties of the canonical map between the prime spectra, orbital and stabilizer algebras, projectivity over the invariant rings, and descent of Cohen-Macaulayness.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.