Construction of self-dual binary [2(2k), 2(2k-1), 2(k)]-codes

The binary Reed-Muller code RM(m - k, m) corresponds to the k-th power of the radical of GF(2)[G], where G is an elementary abelian group of order 2(m) (see [2]). Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for odd m. The group algebra approach enables us to find a self-dual code for even m = 2k in the radical of the previously mentioned group algebra with similarly good parameters as the self-dual RM codes. In the group algebra GF(2)[G] similar or equal to GF(2)[x(1), x(2), ..., x(m)]/(x(1)(2) - 1, x(2)(2) - 1, ... x(m)(2) - 1) we construct self-dual binary C = [2(2k), 2(2k-1), 2(k)] codes with property RM(k - 1, 2k) subset of C subset of RM(k, 2k) for an arbitrary integer k. In some cases these codes can be obtained as the direct product of two copies of RM(k - 1, k)-codes. For k >= 2 the codes constructed are doubly even and for k = 2 we get two non-isomorphic [16, 8, 4]codes. If k > 2 we have some self-dual codes with good parameters which have not been described yet.