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

Crnković (2014) introduced a self-orthogonal [2q,q−12q,q−1] code and a self-dual [2q+2,q+12q+2,q+1] code over the finite field FpFp arising from orbit matrices for Menon designs, for every prime power qq, where q≡1(mod4) and pp a prime dividing q+12. He showed that if qq is a prime and q=12m+5q=12m+5, where mm is a non-negative integer, then the self-dual [2q+2,q+12q+2,q+1] code over F3F3 is equivalent to a Pless symmetry code. However for other values of qq, 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, Crnković (2014) also conjectured that if p=q+12 is a prime, the self-orthogonal code and the self-dual code have minimum distance p+3p+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=25q=25 and p=13p=13.