Numerical experiments related to the covering radius of some first order Reed-Muller codes

Abstract

We have done some numerical experiments to find large coset leaders for the first order Reed-Muller codes RM(m) of length 2^m for odd m,m ≥ 9, by using some matrix groups acting on the field GF(2^m) viewed as a GF(2)-vector space. The coset leaders so obtained are characteristic functions of subsets of GF(2^m) that are union of orbits under the considered group and often have interesting combinatorial properties generalizing difference sets.