MDS Self-dual Codes Research Articles

New MDS Self-Dual Codes Over Finite Field F r 2

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we construct six new classes of MDS self-dual codes by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes are also 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.

MDS or NMDS self-dual codes from twisted generalized Reed–Solomon codes

Self-dual maximum distance separable codes (self-dual MDS codes) and self-dual near MDS (NMDS) codes are very important in coding theory and practice. Thus, it is interesting to construct self-dual MDS or self-dual NMDS codes. In this paper, we not only give parity-check matrices of dual codes of twisted generalized Reed–Solomon codes (TGRS codes) but also present a sufficient and necessary condition of self-dual TGRS codes over $${\mathbb {F}}_q$$ with q an odd prime. Moreover, we construct several classes of self-dual MDS or self-dual NMDS codes from TGRS codes.

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.

A Unified Approach to Construct MDS Self-Dual Codes via Reed-Solomon Codes

MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS selfdual codes over finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> is completely solved for q is even. In the literature, there are many known constructions of MDS self-dual codes for q is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when q is not a square.

New MDS self-dual codes over finite fields of odd characteristic

In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. For odd prime power $q$ with $q$ square, the total number of lengths for MDS self-dual codes over $\mathbb{F}_q$ presented in this paper is much more than those in all the previous results.

Explicit Constructions of MDS Self-Dual Codes

In this paper, we study self-dual codes over finite fields using tools from algebraic function fields in one variable. An algebraic geometry code of length n is defined using two divisors G and D = P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + ⋯ + P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . We characterize self-orthogonality of the genus zero code in terms of the divisors G, D and the value of a well-chosen derivative polynomial at points (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1≤i≤n</sub> . We explore the existence problem of MDS self-dual codes in the odd characteristic case, and we explicitly construct families of new MDS self-dual codes.

Enumeration of Self-Dual Codes of Length 6 over ℤp

The purpose of this paper is to classify and enumerate self-dual codes of length 6 over finite field Z p . First, we classify these codes into three cases: decomposable, indecomposable non-MDS and MDS codes. Then, we complete the classification of non-MDS self-dual codes of length 6 over Z p for all primes p in terms of their automorphism group. We obtain all inequivalent classes and find the necessary and sufficient conditions for the existence of each class. Finally, we obtain the number of MDS self-dual codes of length 6.

Construction of MDS Self-dual Codes over Finite Fields

In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For finite field with odd characteristic and square cardinality, our results can produce more classes of MDS self-dual codes than previous works.

Self-dual codes and orthogonal matrices over large finite fields

In this paper, we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, code extension and projection over a self-dual basis are the main tools. Some optimal, almost MDS, and MDS self-dual codes over both small and large finite fields are constructed. Moreover, over fifty MDS codes with new parameters are constructed. Comparisons with classical constructions are made.

New MDS Self-Dual Codes From Generalized Reed—Solomon Codes

Both MDS and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining existence of $q$-ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where $q$ is even. The current paper focuses on the case where $q$ is odd. We construct a few classes of new MDS self-dual code through generalized Reed-Solomon codes. More precisely, we show that for any given even length $n$ we have a $q$-ary MDS code as long as $q\equiv1\bmod{4}$ and $q$ is sufficiently large (say $q\ge 2^n\times n^2)$. Furthermore, we prove that there exists a $q$-ary MDS self-dual code of length $n$ if $q=r^2$ and $n$ satisfies one of the three conditions: (i) $n\le r$ and $n$ is even; (ii) $q$ is odd and $n-1$ is an odd divisor of $q-1$; (iii) $r\equiv3\mod{4}$ and $n=2tr$ for any $t\le (r-1)/2$.

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.

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 neg...

MDS self-dual codes of lengths 16 and 18

We demonstrate that an MDS self-dual [16, 8, 9] code over double struck capital F p exists for an odd prime number p with 23 ≤ p ≤ 499 by constructing new MDS self-dual codes for the open cases. These codes are obtained using two construction methods based on negacirculant matrices. MDS self-dual [18, 9, 10] codes over double struck capital F p are also constructed for p = 53, 61, 73, 89, 97.

Constructions of self-dual codes over finite commutative chain rings

We study self-dual codes over chain rings. We describe a technique for constructing new self-dual codes from existing codes and we prove that for chain rings containing an element c with c² = −1, all self-dual codes can be constructed by this technique. We extend this construction to self-dual codes over principal ideal rings via the Chinese Remainder Theorem. We use torsion codes to describe the structure of self-dual codes over chain rings and to set bounds on their minimum Hamming weight. Interestingly, we find the first examples of MDS self-dual codes of lengths 6 and 8 and near-MDS self-dual codes of length 10 over a certain chain ring which is not a Galois ring.

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 $${\mathbb{Z}_{p^e}}$$ ). 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.

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.

Euclidean and Hermitian self-dual MDS codes over large finite fields

The first author constructed new extremal binary self-dual codes (IEEE Trans. Inform. Theory 47 (2001) 386–393) and new self-dual codes over GF(4) with the highest known minimum weights (IEEE Trans. Inform. Theory 47 (2001) 1575–1580). The method used was to build self-dual codes from a given self-dual code of a smaller length. In this paper, we develop a complete generalization of this method for the Euclidean and Hermitian self-dual codes over finite fields GF( q). Using this method we construct many Euclidean and Hermitian self-dual MDS (or near MDS) codes of length up to 12 over various finite fields GF( q), where q=8,9,16,25,32,41,49,53,64,81, and 128. Our results on the minimum weights of (near) MDS self-dual codes over large fields give a better bound than the Pless–Pierce bound obtained from a modified Gilbert–Varshamov bound.

MDS Self-Dual Codes over Large Prime Fields

Combinatorial designs have been used widely in the construction of self-dual codes. Recently a new method of constructing self-dual codes was established using orthogonal designs. This method has led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we generalize this method by using generalized orthogonal designs, and we give another new method that creates and solves Diophantine equations over GF(p) in order to find suitable generator matrices for self-dual codes. We show that under the necessary conditions these methods can be applied as well to small and large fields. We apply these two methods to study self-dual codes over GF(31) and GF(37). Using these methods we obtain some new maximum distance separable self-dual codes of small orders.

MDS Self-Dual Codes over Large Prime Fields

Combinatorial designs have been used widely in the construction of self-dual codes. Recently a new method of constructing self-dual codes was established using orthogonal designs. This method has led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we generalize this method by using generalized orthogonal designs, and we give another new method that creates and solves Diophantine equations over GF( p) in order to find suitable generator matrices for self-dual codes. We show that under the necessary conditions these methods can be applied as well to small and large fields. We apply these two methods to study self-dual codes over GF(31) and GF(37). Using these methods we obtain some new maximum distance separable self-dual codes of small orders.