AN EFFICIENT CONSTRUCTION OF SELF-DUAL CODES

Abstract

Abstract. Self-dual codes have been actively studied because of theirconnections with other mathematical areas including t-designs, invari-ant theory, group theory, lattices, and modular forms. We presentedthe building-up construction for self-dual codes over GF(q) with q ≡ 1(mod 4), and over other certain rings (see [19], [20]). Since then, the ex-istence of the building-up construction for the open case over GF(q) withq= p r ≡3 (mod 4) with an odd prime p satisfying p≡3 (mod 4) withr odd has not been solved. In this paper, we answer it positively by pre-senting the building-up construction explicitly. As examples, we presentnew optimal self-dual [16,8,7] codes over GF(7) and new self-dual codesover GF(7) with the best known parameters [24,12,9]. 1. IntroductionSince the development of Algebraic Coding Theory, self-dual codes havebecome one of the main research topics because of their connections withgroups, combinatorial t-designs, lattices, and modular forms (see [26]). Somewell known constructions of self-dual codes include the gluing vector technique([23, 24]) and automorphism group method [15].A recently developed and popular construction is to obtain self-dual codesfrom self-dual codes ofsmaller lengths. In [4, 6], the authors used shadowcodes.Motivated by Harada’s work [12], the second author Kim [17] introduced theso-called building-up construction for binary self-dual codes. It shows that anybinary self-dual code can be built from a self-dual code of a smaller length.Then later, the building-up construction for self-dual codes over finite fieldsGF(q) was developed when q is a power of 2 or q ≡ 1 (mod 4) [19], and thenover finite ring Z