Further Results on the Covering Radii of the Reed-Muller Codes

Let R(r, m) by the rth order Reed-Muller code of length 2 )n, and let o(r, m) be its covering radius. We obtain the following new results on the covering radius of R(r, m): 1. p(r + 1, m + 2) > 2p(r, m) + 2 if 0 < r < m 2. This improves tile successive use of the known inequalities p(r + 1, m + 2) _> 20 (r + 1, m + 1) and p (r + 1, m + 1) -> O (r, m). 2. P (2, 7) -< 44. Previously best known upper bound for p (2, 7) was 46. 3. The covering radius of R(1, m) in R(m I, m) is the same as the covering radius of R(1, m) in R(m 2, m) form > 4.