A Lower Bound on the Optimum Distance Profiles of the Second-Order Reed–Muller Codes

In this paper, we give a lower bound for the optimum distance profiles of the second-order Reed–Muller code in the dictionary order and in the inverse dictionary order. In particular, we investigate the second-order Reed–Muller codes of length
${\leq}256$
. We show that the bound is tight in both orders for the codes of length
${\leq}128$
.