An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

Abstract

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F 2 + uF 2 and F 4 + uF 4 ; using groups of order 3, 7, 9, 13, and 15. Through these constructions and their extensions, we find binary self-dual codes of lengths 32, 40, * This author's visit was partially supported by the London Mathematical Society (International Short Visits -Scheme 5).† Corresponding author 56, 64, 68 and 80, all of which are extremal or optimal. In particular, we find five new self-dual codes of parameters [56, 28,10], twenty-three extremal binary self-dual codes of length 68 with new weight enumerators and fifteen new self-dual codes of parameters [80, 40,14].

New extremal binary self-dual codes from a Baumert–Hall array

“…Most recently codes with new parameters were constructed in [14] (by a bordered four circulant construction) and [2]. Together with these, codes exist with weight enumerators for β =14, 16,18,20,22,24,25,26,28,29,30,32,36,39,44,46,53,59,60,64 and 74 in W 64,1 and for β =0, 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25 In the following tables, we give extremal binary self-dual codes of length 64 obtained from applying constructions described in Theorem 3.1 and Corollary 3.2 over the ring F 4 + uF 4 . The orthogonality-preserving Gray map φ 1 • ϕ F4+uF4 : (F 4 + uF 4 ) n → F 4n 2 is used in finding the binary codes.…”

New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

“…where α, β ∈ Z. Previously known (α, β) values for weight enumerator W 80 can be found online at [30] (see [14,[16][17][18]20,29,33]).…”

New Extremal binary self-dual codes of length 68 from a novel approach to neighbors

“…This idea has been used extensively in the literature to construct new self-dual codes from an existing one. For some of the works that have used this idea, we can refer to [3], [7], [10] and references therein.…”

“…Many different techniques have been employed in constructing these extremal binary self-dual codes such as constructions over certain rings, constructions through automorphism groups, neighboring constructions, shadows, extensions, etc. [1], [4], [7], [8], [10], [13], [14] are just a sample of the works that contain these ideas and their applications in finding new extremal binary self-dual codes.…”