서강대학교

초록/요약

In "D. Crnkovic: Classes of self-orthogonal or self-dual codes from orbit matrices of Menon-designs, Discrete Mathematics 327, 91-95(2014)", Crnkovic 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≡1 (mod 4) and p a prime dividing (q+1)/2. (1) 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 just remarked that these codes do not belong to some previously known series of codes. (2) In particular, Crnkovic conjectured that the code which is constructed by his new method has the minimum distance p + 3 if p=(q+1)/2 is a prime. In this paper, we prove that this is false and that we describe an equivalence between his self-dual codes and the known codes introduced in Dougherty, et al. [10]. We also disprove the conjecture (2) by giving two counter-examples in the case of a self-orthogonal code and a self-dual code, respectively when q = 25 and p = 13.