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

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• 作者：Dong-Man Heo ^{unearthly@sogang.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Jon-Lark Kim ; ^{jlkim@sogang.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Block design ; Linear codes ; Self-dual codes ; Orbit matrix
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 September 2016
• 年：2016
• 卷：339
• 期：9
• 页码：2209-2214
• 全文大小：354 K

文摘

Crnković (2014) introduced a self-orthogonal [2q,q−1$2q,q-1$] code and a self-dual [2q+2,q+1$2q+2,q+1$] code over the finite field F_p${\mathbb{F}}_p$ arising from orbit matrices for Menon designs, for every prime power q$q$, where $q\equiv 1(mod4)$ and p$p$ a prime dividing $\frac{q+1}{2}$. He showed that if q$q$ is a prime and q=12m+5$q=12m+5$, where m$m$ is a non-negative integer, then the self-dual [2q+2,q+1$2q+2,q+1$] code over F₃${\mathbb{F}}_3$ is equivalent to a Pless symmetry code. However for other values of q$q$, 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, Crnković (2014) also conjectured that if $p=\frac{q+1}{2}$ is a prime, the self-orthogonal code and the self-dual code have minimum distance p+3$p+3$. 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$q=25$ and p=13$p=13$.