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 m1, 2(m + 1)/2] is affine-invariant and selfdual, 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 >i 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 rm over GF(2r), and the tools we apply later to the binary codes. © 1994 Academic Press, Inc.