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

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