On self-dual double negacirculant codes

Abstract

Double negacirculant (DN) codes are the analogues in odd characteristic of double circulant codes. Self-dual DN codes of odd dimension are shown to be consta-dihedral. Exact counting formulae are derived for DN codes. The special class of length a power of two is studied by means of Dickson polynomials, and is shown to contain families of codes with relative distances satisfying a modified Gilbert-Varshamov bound.

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.…”

On complementary dual multinegacirculant codes

“…In Proof. Let ξ be a primitive (2n) th root of unity, and assume that x n + 1 factors as in (1). Constituents of C are described in (3).…”

“…Note that the number of linear codes of length 2 over some F Q admitting, along with their dual, a systematic form is Q − 1, all of dimension 1. We are thus excluding the code [1,0] , of dual [0, 1] .…”

On Self-Dual Four Circulant Codes

“…In the present paper, we study index 4 quasi-cyclic codes of parameters [4n, 2n] inspired by [1,3,8]. When x n − 1 has only two irreducible factors, we derive an enumeration formula of the self-dual four circulant codes over F q based on that decomposition, and derive an asymptotic lower bound on the minimum distance of these four circulant codes.…”