Further results on the covering radii of the Reed-Muller codes

Abstract

LetR(r, m) by therth order Reed-Muller code of length2 ^m, and let ρ(r, m) be its covering radius. We obtain the following new results on the covering radius ofR(r, m): 1. ρ(r+1,m+2)≥ 2ρ(r, m)+2 if 0≤r≤m−2. This improves the successive use of the known inequalities ρ(r+1,m+2)≥2ρ(r+1,m+1) and ρ(r+1,m+1)≥ρ (r, m).2.ρ(2, 7)≤44. Previously best known upper bound for ρ(2, 7) was 46. 3. The covering radius ofR(1,m) inR(m−1,m) is the same as the covering radius ofR(1,m) inR(m−2,m) form≥4.