Projected subcodes of the second order binary Reed-Muller code

Abstract

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.