Formally self-dual additive codes over F4 and near-extremal self-dual codes

We introduce a class of formally self-dual additive codes over F4 as a natural analogue of binary formally self-dual codes, which is missing in the study of additive codes over F4. We define extremal formally self-dual additive codes over F4 and classify all such codes. Interestingly, we find exactly three formally self-dual additive (7, 27) odd codes over F4 with minimum distance d = 4, a better minimum distance than any self-dual additive (7, 27) codes over F4. We further define near-extremal formally selfdual additive codes over F4 as an analogue of near-extremal binary formally self-dual codes, and prove that they do not exist if their lengths are n = 16, 18 or n ≥ 20. We improve the bounds on the minimum distance of formally self-dual binary codes in a similar manner. Finally, we extend S. Zhang’s best known upper bound on the highest minimum distance of the four types of classical self-dual codes of large lengths.