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.     

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.     

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.     

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.     

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.     

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.     

The depth spectrums of constacyclic codes over finite chain rings

In light of the generator polynomials of constacyclic codes over finite chain rings, the depth spectrum of constacyclic codes can be determined if (n,p)=1$(n,p)=1$.     

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.     

Polynomial orbits in finite commutative rings

Let R be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.     

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     

Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with $p^4$ elements, (p a prime) are given. Second, the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined.     

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.