71 research outputs found
New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2
In this work, new construction methods for self-dual codes are given. The
methods use the short Kharaghani array and a variation of it. These are
applicable to any commutative Frobenius ring. We apply the constructions over
the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various
weight enumerators obtained as Gray images. By the use of an extension theorem
for self-dual codes we were able to construct 27 new extremal binary self-dual
codes of length 68. The existence of the extremal binary self-dual codes with
these weight enumerators was previously unknown.Comment: 10 pages, 5 table
New extremal singly even self-dual codes of lengths and
For lengths and , we construct extremal singly even self-dual codes
with weight enumerators for which no extremal singly even self-dual codes were
previously known to exist. We also construct new inequivalent extremal
doubly even self-dual codes with covering radius meeting the
Delsarte bound.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1706.0169
Modified Quadratic Residue Constructions and New Exermal Binary Self-Dual Codes of Lengths 64, 66 and 68
In this work we consider modiļ¬ed versions of quadratic double circulant and quadratic bordered double circulant constructions over the binary ļ¬eld and the rings F2 +uF2 and F4 +uF4 for diļ¬erent prime values of p. Using these constructions with extensions and neighbors we are able to construct a number of extremal binary self-dual codes of diļ¬erent lengths with new parameters in their weight enumerators. In particular we construct 2 new codes of length 64, 4 new codes of length 66 and 14 new codes of length 68. The binary generator matrices of the new codes are available online at [8]
New binary self-dual codes via a generalization of the four circulant construction
In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets , , , we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.https://www.mathos.unios.hr/mc/index.php/mc/article/view/352
New binary self-dual codes via a variation of the four-circulant construction
In this work, we generalize the four circulant construction for self-dual
codes. By applying the constructions over the alphabets F_2, F_2+uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68.
More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed
2^n Bordered Constructions of Self-Dual codes from Group Rings
Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes.
Various techniques involving circulant matrices and matrices from group rings have been used to construct
such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary
self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual
codes by combining many of the previously used techniques. The purpose of this is to construct self-dual
codes that were missed using classical construction techniques by constructing self-dual codes with diļ¬erent
automorphism groups. We apply the technique to codes over ļ¬nite commutative Frobenius rings of characteristic
2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct
some extremal self-dual codes length 64 and 68, constructing 30 new extremal self-dual codes of length 68
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
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