Let $X$ be a topological space. Cohomology operations are generated by the natural transformations of degree $i$ which are so-called Steenrod squares $Sq^{i}:H^{*}(X,\mathbb{F}_{2})\longrightarrow H^{*+i}(X,\mathbb{F}_{2}),$ where $H^{*}(X, \mathbb{F}_{2})$ is the singular cohomology of $X$ with coefficients in the two-element field $ \mathbb{F}_{2},$ and $i$ is arbitrary non-negative integers. The algebra of stable cohomology operations with coefficients in $ \mathbb{F}_{2}$ is known as the modulo 2 Steenrod algebra, $ \mathcal{A}.$ Let $\mathcal P_n:=H^{*}\big(BE_n; \mathbb F_2 \big) \cong \mathbb F_2[x_{1},x_{2},\ldots,x_{n}]$ be the graded polynomial algebra over the prime field of two elements $\mathbb F_2$, in $n$ generators $x_1, x_2, \ldots , x_n$, each of degree 1, where $E_n$ is an elementary abelian 2-group of rank $n,$ and $BE_n$ is the classifying space of $E_n.$ We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\mathcal{A}$. This paper aims to explicitly determine an admissible monomial basis of the $\mathbb{F}_{2}$-vector space $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{6}$ in the generic degree $2^{s+3}-6,$ with $s$ an arbitrary non-negative integer. As an application of the above results, we obtain the dimension results for the polynomial algebra $\mathcal P_n$ in degree $d= (n-1).(2^{n+u-1}-1)+\ell.2^{n+u},$ where $u$ is an arbitrary non-negative integer, $\ell =13,$ and $n=7.$
