The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees

Let $P_k$ be the graded polynomial algebra ${\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$ 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, $\mathcal{A}\mathrm{.}$ 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 $d⩾6$.