  Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$
with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the
prime field with two elements. The hit problem of Frank Peterson asks for a
minimal generating set for the polynomial algebra $P_k$ as a module over the
mod-2 Steenrod algebra $\mathcal{A}$. Equivalently, we want to find a vector
space basis for $\mathbb F_2 \otimes_{\mathcal A} P_k$ in each degree. In this
paper, we study a generating set for the kernel of Kameko's squaring operation
$\widetilde{Sq}^0_*: \mathbb F_2 \otimes_{\mathcal A} P_k \longrightarrow
\mathbb F_2 \otimes_{\mathcal A} P_k$ in a so-called generic degree. By using
this result, we explicitly compute the hit problem for $k=5$ in the respective
generic degree.
