Duals of constacyclic codes over finite local Frobenius non-chain rings of length 4

Duals of constacyclic codes over a finite local Frobenius non-chain ring of length 4, the length of which is relatively prime to the characteristic of the residue field of the ring are determined. Generators for the dual code are obtained from those of the original constacyclic code. In some cases self-dual codes are determined.     

The dual of a constacyclic code,self dual,reversible constacyclic codes and constacyclic codes with complementary duals over finite local Frobenius non-chain rings with nilpotency index 3

Over finite local Frobenius non-chain rings with nilpotency index 3 and when the length of the codes is relatively prime to the characteristic of the residue field of the ring, the structure of the dual of $\gamma$-constacyclic codes is established and the algebraic characterization of self-dual $\gamma$-constacyclic codes, reversible $\gamma$-constacyclic codes and $\gamma$-constacyclic codes with complementary dual are given. Generators for the dual code are obtained from those of the original constacyclic code.     

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.     

Additive cyclic codes over finite commutative chain rings

Additive cyclic codes over Galois rings were investigated in Cao et al. (2015). In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the study in Cao et al. (2015), whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.     

Contraction of cyclic codes over finite chain rings

In this paper, $R$ is a finite chain ring with residue field ${\mathbb{F}}_q$ and $\gamma$ is a unit in $R.$ By assuming that the multiplicative order $u$ of $\gamma$ is coprime to $q,$ we give the trace-representation of any simple-root $\gamma$-constacyclic code over $R$ of length $?,$ and on the other hand show that any cyclic code over $R$ of length $u?$ is a direct sum of trace-representable cyclic codes. Finally, we characterize the simple-root, contractable and cyclic codes over $R$ of length $u?$ into $\gamma$-constacyclic codes of length $?.$     

On quantum stabilizer codes derived from local Frobenius rings

In this paper we consider stabilizer codes over local Frobenius rings. Firstly, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on the isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems.     

The linear programming bound for codes over finite Frobenius rings

In traditional algebraic coding theory the linear-programming bound is one of the most powerful and restrictive bounds for the existence of both linear and non-linear codes. This article develops a linear-programming bound for block codes on finite Frobenius rings. An erratum to this article can be found at     

MDS codes over finite principal ideal rings

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 ). 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.      

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

On the equivalence of codes over rings and modules

In light of the result by Wood that codes over every finite Frobenius ring satisfy a version of the MacWilliams equivalence theorem, a proof for the converse is considered. A strategy is proposed that would reduce the question to problems dealing only with matrices over finite fields. Using this strategy, it is shown, among other things, that any left MacWilliams basic ring is Frobenius. The results and techniques in the paper also apply to related problems dealing with codes over modules.     

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.     

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.