A modified bordered construction for self-dual codes from group rings

Abstract

We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78.

New binary self-dual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction

“…Hexadecimal notation system for elements of F2 + uF2, F2 + uF2 + vF2 + uvF2, F4 and F4 + uF4. [21,14,15,26]) and the existence of codes with weight enumerator W 56,2 has previously been determined for α ∈ {−z : z = 0, 2, 4, ..., 32, 34, 35, 38, 40, 42, 56} (see [21,14,15,26]).…”

New binary self-dual codes of lengths 56, 62, 78, 92 and 94 from a bordered construction

“…where α ∈ Z. Previously known α values for weight enumerators W56,1 and W56,2 can be found online at [34] (see [27,19,20,33,22]).…”

“…where α, β ∈ Z. Previously known (α, β) values for weight enumerator W78,1 can be found online at [34] (see [2,24,16,40,37,20]). We obtain one new extremal binary self-dual codes of length 78 which has weight enumerator W78,1 for β = 0 and α = −76.…”

“…There exist several papers in which such constructions are used to obtain binary self-dual codes of different lengths with new parameters in their weight enumerators. Examples of bordered constructions can be found in [15,14,20]. In this work, we continue in this direction and we present a new bordered matrix construction that we employ to obtain many binary self-dual codes of lengths 56, 62, 78, 92 and 94 with weight enumerators that were not known in the literature before.…”