On self-dual affine-invariant codes

An extended cyclic code of length 2^m over GF(2) cannot be self-dual for even m. For odd m, the Reed-Muller code [2^m, 2^m-1, 2 ^{(m+1) 2}] is affine-invariant and self-dual, and it is the only such code for m = 3 or 5. We describe the set of binary self-dual affine-invariant codes of length 2^m for m = 7 and m = 9. For each odd m, m ≥ 9, we exhibit a self-dual affine-invariant code of length 2^m over GF(2) which is not the self-dual Reed-Muller code. In the first part of the paper, we present the class of self-dual affine-invariant codes of length 2^m over GF(2^r), and the tools we apply later to the binary codes.