Negacyclic self-dual codes over finite chain rings

In this article, we study negacyclic self-dual codes of length n over a finite chain ring R when the characteristic p of the residue field [`(R)]{\bar{R}} and the length n are relatively prime. We give necessary and sufficient conditions for the existence of (nontrivial) negacyclic self-dual codes over a finite chain ring. As an application, we construct negacyclic MDR self-dual codes over GR(p ^t , m) of length p ^m  + 1.     

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 $?.$     

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.     

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let $R$ be the Galois ring $GR(p^k,s)$ of characteristic $p^k$ and cardinality $p^{sk}$. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length $n$ over $R$, and a $p$-adic integer ring with $gcd(p,n)=1$. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length $r{l}^m$ over $R$, where $r,l,$ and $t$ are three primes with $2?l$, $r|(q^t?1)$, $l^v\parallel (q^t?1)$ and $gcd(rl,q(q?1))=1$. Finally, as applications, weight distributions of all irreducible cyclic codes for $t=2$ and generator polynomials of self-dual cyclic codes of length $l^m$ and $r{l}^m$ over $R$ are given.     

有限链环上的常循环码

研究了有限链环R上常循环码的等价性,根据等价性给出了R上一些常循环码及其对偶码的结构.确定了该环上长度为ps的所有常循环码及其对偶码的结构.     

MDS and self-dual codes over rings

In this paper we give the structure of constacyclic codes over formal power series and chain rings. We also present necessary and sufficient conditions on the existence of MDS codes over principal ideal rings. These results allow for the construction of infinite families of MDS self-dual codes over finite chain rings, formal power series and principal ideal rings. We also define the Reed–Solomon codes over principal ideal rings.     

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.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

New MDS self-dual codes over finite fields

In this paper we construct MDS Euclidean and Hermitian self-dual codes which are extended cyclic duadic codes or negacyclic codes. We also construct Euclidean self-dual codes which are extended negacyclic codes. Based on these constructions, a large number of new MDS self-dual codes are given with parameters for which self-dual codes were not previously known to exist.     

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$(n,p)=1$.     

Repeated-root cyclic and negacyclic codes over a finite chain ring

We show that repeated-root cyclic codes over a finite chain ring are in general not principally generated. Repeated-root negacyclic codes are principally generated if the ring is a Galois ring with characteristic a power of 2. For any other finite chain ring they are in general not principally generated. We also prove results on the structure, cardinality and Hamming distance of repeated-root cyclic and negacyclic codes over a finite chain ring.     

Linearly representable codes over chain rings

In this paper, we consider linear codes over finite chain rings. We present a general mapping which produces codes over smaller alphabets. Under special conditions, these codes are linear over a finite field. We introduce the notion of a linearly representable code and prove that certain MacDonald codes are linearly representable. Finally, we give examples for good linear codes over finite fields obtained from special multisets in projective Hjelmslev planes.