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

Carolin Hannusch ; Piroska Lakatos

Algebra and discrete mathematics, Tome 21 (2016) no. 1, pp. 59-68

Résumé

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 [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 $$GF(2)[G]\cong GF(2)[x_1,x_2,\dots, x_m]/(x_1^2-1,x_2^2-1, \dots x_m^2-1)$$ 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.

Détail

Keywords: Reed–Muller code, Generalized Reed–Muller code, radical, self-dual code, group algebra, Jacobson radical.