Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes

Abstract

The binary Reed-Muller code \({\rm RM}(m-k,m)\) 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~\cite{B}). 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
\[ GF(2)[G]\cong GF(2)[x_1,x_2,\dots, x_m]/(x_1^2-1,x_2^2-1, \dots x_m^2-1)\]
we construct self-dual binary \(C=[2^{2k},2^{2k-1},2^k]\) codes with property
\[{\rm RM}(k-1,2k) \subset C \subset {\rm RM}(k,2k)\] for an arbitrary integer \(k\).

In some cases these codes can be obtained as the direct product of two copies of \({\rm RM}(k-1,k)\)-codes. For \(k\geq 2\) 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.