  Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$
over the prime field with 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, $\mathcal{A}.$ It is an open problem in Algebraic Topology. In this
paper, we explicitly determine a minimal set of $\mathcal{A}$-generators for
$P_5$ in the case of the generic degree $m = 2^{d}$ for all $d \geqslant 8$.
