On self-dual codes over $${\mathbb{F}}_5$$

Abstract

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over $${\mathbb{F}}_5$$ for lengths [22, 26, 28, 32---40]. In particular, we prove that there is no [22, 11, 9] self-dual code over $${\mathbb{F}}_5$$ , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over $${\mathbb{F}}_5$$ are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18, 9, 7] codes over $${\mathbb{F}}_5$$ up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over $${\mathbb{F}}_5$$ and at least 40 new inequivalent optimal self-dual [22, 11, 8] codes.