On some double circulant codes of Crnkovic and disproof of his two conjectures

초록

Crnkovic (2014) introduced a self-orthogonal [2q, q - 1] code and a self-dual [2q + 2, q + 1] code over the finite field F-p arising from orbit matrices for Menon designs, for every prime power q, where q 1 (mod 4) and p a prime dividing q+1/2. He showed that if q is a prime and q = 12m +5, where m is a non-negative integer, then the self -dual [2q+2, q + 1] code over F-3 is equivalent to a Pless symmetry code. However for other values of q, he remarked that these codes, up to his knowledge, do not belong to some previously known series of codes. In this paper, we describe an equivalence between his self -dual codes and the known codes introduced by Gaborit in 2002. On the other hand, Cmkovid (2014) also conjectured that if p = 9211 is a prime, the self-orthogonal code and the self -dual code have minimum distance p + 3. We disprove this conjecture by giving two counter-examples in the case of the self-orthogonal code and the self-dual code, respectively when q = 25 and p = 13. (C) 2016 Elsevier B.V. All rights reserved.

키워드