Let $$ be the graded polynomial algebra over the prime field of two elements $$, in $$ generators $$, each of degree 1. Being the mod-2 cohomology of the classifying space $$, the algebra $$ is a module over the mod-2 Steenrod algebra $$.

In this Note, we explicitly compute the hit problem of some generic degrees $$ in $$, where $$ and $$ an arbitrary non-negative integer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees and in the cases $$ and 6.