Some inequalities about the covering radius of Reed-Muller codes

Abstract

Let R(r, m) be the rth order Reed-Muller code of length 2^m, and let ρ(r, m) be its covering radius. We prove that if 2≤ k ≤ m - r - 1, then ρ(r + k, m + k) ≥ ρ(r, m + 2(k - 1). We also prove that if m - r ≥ 4, 2 ≤ k ≤ m - r - 1, and R(r, m) has a coset with minimal weight ρ(r, m) which does not contain any vector of weight ρ(r, m) + 2, then ρ(r + k, m + k) ≥ ρ(r, m) + 2k(. These inequalities improve repeated use of the known result ρ(r + 1, m + 1) ≥ ρ(r, m).