An efficient construction of self-dual codes

摘要

Self-dual codes have been actively studied because of their connections with other mathematical areas including $t$-designs, invariant theory, group theory, lattices, and modular forms. We presented the building-up construction for self-dual codes over $GF(q)$ with $q equiv 1 pmod 4$, and over other certain rings (see~cite{KimLee},~cite{KimLee2}). Since then, the existence of the building-up construction for the open case over $GF(q)$ with $q=p^r equiv 3 pmod 4$ with an odd prime $p$ satisfying $p equiv 3 pmod 4$ with $r$ odd has not been solved. In this paper, we answer it positively by presenting the building-up construction explicitly. As examples, we present new optimal self-dual $[16,8,7]$ codes over $GF(7)$ and new self-dual codes over $GF(7)$ with the best known parameters $[24,12,9]$.