Construction of long MDS self-dual codes from short codes

In this paper, we propose a mechanism on how to construct long MDS self-dual codes from short ones. These codes are special types of generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes. The main tool is utilizing additive structure or multiplicative structure on finite fields. By applying this method, more MDS self-dual codes can be constructed.     

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.     

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.     

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.     

Structure Theorems for Group Ring Codes with an Application to Self-Dual Codes

Using ideas from the cohomology of finite groups, an isomorphism is established between a group ring and the direct sum of twisted group rings. This gives a decomposition of a group ring code into twisted group ring codes. In the abelian case the twisted group ring codes are (multi-dimensional) constacyclic codes. We use the decomposition to prove that, with respect to the Euclidean inner product, there are no self-dual group ring codes when the group is the direct product of a 2-group and a group of odd order, and the ring is a field of odd characteristic or a certain modular ring. In particular, there are no self-dual abelian codes over the rings indicated. Extensions of these results to non-Euclidean inner products are briefly discussed.     

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.     

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.      

有限域上长为2mpn 的负循环码

设F_q 是奇数阶有限域. 本文主要借助X^2mpn+1 在F_q 上的不可约因式分解来确定有限域F_q上所有长为2^mpⁿ 的负循环码和自对偶的负循环码的生成多项式, 这里p 是q-1 的奇素因子, m 和n是正整数.     

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.      

SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.     

A complete classification of ternary self-dual codes of length 24

Ternary self-dual codes have been classified for lengths up to 20. At length 24, a classification of only extremal self-dual codes is known. In this paper, we give a complete classification of ternary self-dual codes of length 24 using the classification of 24-dimensional odd unimodular lattices.     

Binary Optimal Odd Formally Self-Dual Codes

In this paper, we study binary optimal odd formallyself-dual codes. All optimal odd formally self-dual codes areclassified for length up to 16. The highest minimum weight ofany odd formally self-dual codes of length up to 24 is determined. We also show that there is a unique linearcode for parameters [16, 8, 5] and [22, 11, 7], up to equivalence.     

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.     

ON SELF-DUAL PERMUTATION CODES

Permutation codes over finite fields are introduced, some conditions for exis-tence or non-existence of self-dual permutation codes are obtained.     

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.     

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.     

Codes Over the p-adic Integers

We study codes over the p-adic integers and correct errors in the existing literature. We show that MDS codes exist over the p-adics for all lengths, ranks and p. We show that self-dual codes exist over the 2-adics if and only if the length is a multiple of 8 and that self-dual codes exist over the p-adics with p odd if and only if the length is 0 (mod 4) for p ≡ 3 (mod 4) and 0 (mod 2) for p ≡ 1 (mod 4).