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

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over for lengths 22, 26, 28, 32–40]. In particular, we prove that there is no 22, 11, 9] self-dual code over , 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 are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual 18, 9, 7] codes over up to the monomial equivalence, and construct one new optimal self-dual 20, 10, 8] code over and at least 40 new inequivalent optimal self-dual 22, 11, 8] codes.