On the norm and covering radius of the first-order Reed-Muller codes

Let rho(1, m) and N(1, m) be the covering radius and norm of the first-order Reed-Muller code R(1, m), respectively. It is known that rho(1, 2k + 1) less than or equal to [2(2k) - 2((2k - 1)/2)] and N(1, 2k + 1) less than or equal to 2[2(2k) - 2((2k - 1)/2)] (k > 0). We prove that rho(1, 2k + 1) less than or equal to 2[2(2k - 1) - 2((2k - 3)/2)] and N(1, 2k + 1) less than or equal to 4[2(2k - 1) - 2((2k - 3)/2)] ( k > 0). We also discuss the connections of the two new bounds with other coding theoretic problems.