  Denote by $P_k$ 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 Peterson hit problem of determining 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 $P_k$ in the case $k=5$ and the degree $4(2^d -
1)$ with $d$ an arbitrary positive integer.
