Shadow Bounds for Self-Dual Codes

Conway and Sloane have previously given an upper bound on the minimum distance of a singly-even self-dual binary code, using the concept of the shadow of a self-dual code. We improve their bound, finding that the minimum distance of a selfdual binary code of length n is at most 4bn=24c+4, except when nmod24 = 22 , when the bound is 4bn=24c + 6. We also show that a code of length a multiple of 24 meeting the bound cannot be singly-even. The same technique gives similar results for additive codes over GF (4) (relevant to quantum coding theory).