  We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t =
\mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1,
\ldots, x_t$ with the degree of each $x_i$ being one. Let $\mathcal A_2$ be the
Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to
determine a minimal set of generators for the "cohits" space $\mathbb
Z_2\otimes_{\mathcal A_2} P_t.$ This problem, which is called the "hit" problem
for Steenrod algebra, has been systematically studied for $t\leq 4.$ The
present paper is devoted to the investigation of the structure of $\mathbb
Z_2\otimes_{\mathcal A_2} P_t$ in some certain "generic" degrees. More
specifically, we explicitly determine a monomial basis of $\mathbb
Z_2\otimes_{\mathcal A_2} P_5$ in degree $n_s=5(2^{s}-1) + 42.2^{s}$ for every
non-negative integer $s.$ As a result, it confirms Sum's conjecture [14] for a
relation between the minimal sets of $\mathcal A_2$-generators of the algebras
$P_{t-1}$ and $P_{t}$ in the case $t=5$ and degree $n_s$. As applications, we
obtain the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_6$ in the generic
degree $5(2^{s+5}-1) + n_0.2^{s+5}$ for all $s\geq 0,$ and show that the
Singer's cohomological transfer [11] is an isomorphism in bidegree $(5,
5+n_s)$.
