Some results on the covering radii of Reed-Muller codes

Let R(r,m) be the rth-order Reed-Muller code of length 2^m and let ρ(r,m ) be its covering radius. R(2,7), R(2,8), R (3,7), and R(4,8) are among those smallest Reed-Muller codes whose covering radii are not known. New bounds for the covering radii of these four codes are obtained. The results are ρ(2,7)⩾40, ρ(2,8)⩾84, 20⩽ρ(3,7)⩽23, and ρ(4,8)⩾22. Noncomputer proofs for the known results that ρ(2,6)=18 and that R(1,5) is normal are given