Construction of quasi-cyclic self-dual codes

Abstract

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field ${\mathbb{F}}_q$ and linear codes over a ring $R={\mathbb{F}}_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field ${\mathbb{F}}_q$ can be obtained by the {\it building-up} construction, provided that char $({\mathbb{F}}_q)=2$ or $q\equiv 1(\mathrm{mod}4)$, $m$ is a prime $p$, and $q$ is a primitive element of ${\mathbb{F}}_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ a prime) in terms of divisibility by $p$. We improve the result of [3] by constructing new binary cubic (i.e., $\ell$-quasi-cyclic codes of length $3\ell$) optimal self-dual codes of lengths $30,36,42,48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40,20,12]$ code over ${\mathbb{F}}_3$ and a new 6-quasi-cyclic self-dual $[30,15,10]$ code over ${\mathbb{F}}_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28,14,9]$ code over ${\mathbb{F}}_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over ${\mathbb{F}}_4$.

arXiv comment25 pages, 2 tables; Finite Fields and Their Applications, 2012