  Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$
over the prime field of two elements, $\mathbb F_2$, with the degree of each
$x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a
minimal set of generators for $P_k$ as a module over the mod-$2$ Steenrod
algebra, $\mathcal{A}.$ In this paper, we explicitly determine a minimal set of
$\mathcal{A}$-generators for $P_5$ in the case of the degrees $n = 2^{d+1} - 1$
and $n = 2^{d+1} - 2$ for all $d \geqslant 6$.
