103 research outputs found
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
Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31
A classification of Hadamard matrices of order with an automorphism of
order is given for and . The ternary self-dual codes spanned by
the newly found Hadamard matrices of order with an automorphism of order
are computed, as well as the binary doubly even self-dual codes of length
with generator matrices defined by related Hadamard designs. Several new
ternary near-extremal self-dual codes, as well as binary near-extremal doubly
even self-dual codes with previously unknown weight enumerators are found.Comment: 21 page
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
Ternary extremal four-negacirculant self-dual codes
In this note, we give basic properties of ternary four-negacirculant
self-dual codes. By exhaustive computer search based on the properties, we
complete a classification of ternary extremal four-negacirculant self-dual
codes of lengths 40, 44, 48, 52 and 60.Comment: arXiv admin note: text overlap with arXiv:2303.0505
New examples of self-dual near-extremal ternary codes of length 48 derived from 2-(47,23,11) designs
In a recent paper [M. Araya, M. Harada, Some restrictions on the weight
enumerators of near-extremal ternary self-dual codes and quaternary Hermitian
self-dual codes, Des. Codes Cryptogr., 91 (2023), 1813--1843], Araya and Harada
gave examples of self-dual near-extremal ternary codes of length 48 for
distinct values of the number of codewords of minimum weight 12, and
raised the question about the existence of codes for other values of .
In this note, we use symmetric 2- designs with an automorphism
group of order 6 to construct self-dual near-extremal ternary codes of length
48 for new values of .Comment: 7 page
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