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.      

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.     

Skew quasi-cyclic codes over Galois rings

Recently there has been a lot of interest on algebraic codes in the setting of skew polynomial rings. In this paper we have studied skew quasi-cyclic (QC) codes over Galois rings. We have given a necessary and sufficient condition for skew cyclic codes over Galois rings to be free, and determined a distance bound for free skew cyclic codes. A sufficient condition for 1-generator skew QC codes to be free is determined. Some distance bounds for free 1-generator skew QC codes are discussed. A canonical decomposition of skew QC codes is presented.     

Galois LCD codes over finite fields

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k-Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k-Galois LCD codes is obtained, two classes of k-Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ-constacyclic codes being k-Galois LCD codes are characterized. Some classes of k-Galois LCD λ-constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ-constacyclic codes, and present a class of Hermitian LCD λ-constacyclic MDS codes.     

New entanglement-assisted quantum codes from k-Galois dual codes

Entanglement-assisted quantum error-correcting (EAQEC, for short) codes use pre-existing entanglements between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, they can only be constructed from classical linear codes which contain their Hermitian dual codes. However, how to determine the parameters of ebits c in EAQEC codes is not an easy task. In this paper, let p be prime and e, k be integers, we construct six classes of EAQEC codes based on k-Galois dual codes over finite fields ${\mathbb{F}}_{p^e}$, where $0\le k<e$. The parameter of ebits c of these EAQEC codes can be easily generated algebraically. Furthermore, the six classes of EAQEC codes are of maximal entanglement, most of which have better parameters than current EAQEC codes available.     

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.     

Fq-linear codes over Fq-algebras

Huffman (2013) [12] studied ${\mathbb{F}}_q$-linear codes over ${\mathbb{F}}_{q^m}$ and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative ${\mathbb{F}}_q$-algebra. An ${\mathbb{F}}_q$-linear code over S of length n is an ${\mathbb{F}}_q$-submodule of $S^n$. In this paper, we study ${\mathbb{F}}_q$-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over ${\mathbb{F}}_q$-algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of ${\mathbb{F}}_q$-linear codes over finite commutative graded ${\mathbb{F}}_q$-algebras.     

Negacyclic codes over Galois rings of characteristic 2 a

We investigate negacyclic codes over the Galois ring GR(2 a ,m) of length N = 2 k n,where n is odd and k 0.We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR(2 a ,m)[u]/ u 2 k + 1 .Then using a ring isomorphism we obtain the structure of negacyclic codes over GR(2 a ,m) of length N = 2 k n (n odd) and explore the existence of self-dual negacyclic codes over GR(2 a ,m).A bound for the homogeneous distance of such negacyclic codes is also given.     

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 $?.$     

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.     

Galois self-dual extended duadic constacyclic codes

Galois inner product is a generalization of the Euclidean inner product and Hermitian inner product. The theory on linear codes under Galois inner product can be applied in the constructions of MDS codes and quantum error-correcting codes. In this paper, we construct Galois self-dual codes and MDS Galois self-dual codes from extensions of constacyclic codes. First, we explicitly determine all the Type II splittings leading to all the Type II duadic constacyclic codes in two cases. Second, we propose methods to extend two classes of constacyclic codes to obtain Galois self-dual codes, and we also provide existence conditions of Galois self-dual codes which are extensions of constacyclic codes. Finally, we construct some (almost) MDS Galois self-dual codes using the above results. Some Galois self-dual codes and (almost) MDS Galois self-dual codes obtained in this paper turn out to be new.     

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.      

New Galois hulls of generalized Reed-Solomon codes

Most recently, Gao et al. found a nice method to investigate the Euclidean hulls of generalized Reed-Solomon codes in terms of Goppa codes. In this note, we extend the results to general Galois hull. We prove that the Galois hulls of some GRS codes are still GRS codes. We also give some examples on Galois LCD and self-dual MDS codes. Compare with known results, the Galois hulls of GRS codes obtained in this work have flexible parameters.     

Matrix graphs and MRD codes over finite principal ideal rings

Let R be a finite principal ideal ring and $m,n,d$ positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are $m\times n$ matrices over R and two matrices A and B are adjacent if and only if $0<\mathrm{rank}(A-B)<d$. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R.