Self-dual codes and orthogonal matrices over large finite fields

Shi, Minjia; Sok, Lin; Solé, Patrick; Çalkavur, Selda

doi:10.1016/j.ffa.2018.08.011

Cited by 32 publications

(7 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An element of C is called a codeword. A generator matrix of C is a matrix whose rows form a basis of C. For vectors x = (x i ) and y = (y i ), we define the inner product [7], [13], [14], [16], [17], [25], [36]- [38]. New results from this article written in bold.…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Self-Dual Codes, Symmetric Matrices, and Eigenvectors

Kim

Choi

2021

IEEE Access

View full text Add to dashboard Cite

We introduce a consistent and efficient method to construct self-dual codes over GF (q) using symmetric matrices and eigenvectors from a self-dual code over GF (q) of smaller length where q ≡ 1 (mod 4). Using this method, which is called a 'symmetric building-up' construction, we improve the bounds of the best-known minimum weights of self-dual codes with lengths up to 40, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, which includes double circulant codes. We obtain 2967 new self-dual codes over GF (13) and GF (17) up to equivalence. Also, we compute the minimum weights of quadratic residue(QR) codes that were previously unknown. These are: [20,10,10] QR self-dual code over GF (23), [24,12,12] QR self-dual codes over GF (29) and GF (41), and [32,16,14] QR self-dual codes over GF (19). They have the highest minimum weights so far.INDEX TERMS eigenvectors, optimal codes, quadratic residue codes, self-dual codes, symmetric matrix, symmetric self-dual code

show abstract

Section: Preliminariesmentioning

confidence: 99%

“…In Section 3, we present a construction method of symmetric TABLE 2. The best-known minimum weights of self-dual codes of length n over GF (q) where n ≤ 40 and 5 ≤ q ≤ 19 [3], [7], [12], [14], [15], [18], [20], [27], [36]. New results from this article written in bold.…”

Section: Introductionmentioning

confidence: 99%

Self-Dual Codes, Symmetric Matrices, and Eigenvectors

Kim

Choi

2021

IEEE Access

View full text Add to dashboard Cite

show abstract

“…With the most updated information, the existence of codes is known for β =14, 18,22,25,29,32,35,36,39,44,46,53,59, 60, 64 and 74 in W 64,1 and for β =0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, . .…”

Section: Constructions Coming From Dmentioning

confidence: 99%

Double bordered constructions of self-dual codes from group rings over Frobenius rings

et al. 2020

View full text Add to dashboard Cite

In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings F 2 + uF 2 and F 4 + uF 4. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables. Keywords Group rings • Self-dual codes • Codes over rings • Extremal codes • Bordered constructions Mathematics Subject Classification (2010) 94B05 • 94B15 This research was supported by the London Mathematical Society (International Short Visits-Scheme 5).

show abstract

“…From now on, we consider q = p m where p is an odd prime number and m ≥ 1. Several families of MDS self-dual codes over F q have been constructed with length n satisfying certain conditions by using generalized Reed-Solomon (GRS for short) codes and extended generalized Reed-Solomon (EGRS for short) codes [2]- [4], [7], [9]- [12], [15], orthogonal designs [8,14], extended cyclic duadic codes and negacyclic codes [6]. Roughly speaking, the first approach is to look for the GRS codes and EGRS codes as candidates of MDS codes, then to find sufficient conditions satisfied by length n such that the codes are self-dual.…”

Section: Introductionmentioning

confidence: 99%

“…The last two approaches are to look for the self-dual codes given by orthogonal designs and (nega-)cyclic codes and select ones being MDS codes. A table of MDS self-dual codes over F q is provided in [14] for length n ≤ 12 and odd prime number p ≤ 109.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

Feng

2020

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

Based on the fundamental results on MDS self-dual codes over finite fields constructed via generalized Reed-Solomon codes [9] and extended generalized Reed-Solomon codes [15], many series of MDS self-dual codes with different length have been obtained recently by a variety of constructions and individual computations. In this paper, we present an unified approach to get several previous results with concise statements and simplified proofs, and some new constructions on MDS self-dual codes. In the conclusion section we raise two open problems.2010 Mathematics Subject Classification. 11T06, 11T55.

show abstract

Self-dual codes and orthogonal matrices over large finite fields

Cited by 32 publications

References 16 publications

Self-Dual Codes, Symmetric Matrices, and Eigenvectors

Self-Dual Codes, Symmetric Matrices, and Eigenvectors

Double bordered constructions of self-dual codes from group rings over Frobenius rings

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

Contact Info

Product

Resources

About