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

Let ρ(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 ρ(1,2k+1)⩽lower bound [2^2k-2^(2k-1/2)] and N(1,2k+1)⩽2 lower bound [2^2k-2^(2k-1/2)] (k>0). We prove that ρ(1,2k+1)⩽2 lower bound [2^2k-1-2^(2k-3/2)] and N(1,2k+1)⩽4 lower bound [2^2k-1-2^(2k-3/2)] (k>0). We also discuss the connections of the two new bounds with other coding theoretic problems