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 \(\mathbb F_2[x_1,x_2,\ldots ,x_k]\) 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 \(d \geqslant 6\).