  Let $P_k:= \mathbb{F}_2[x_1,x_2,\ldots ,x_k]$ be the polynomial algebra in
$k$ variables with the degree of each $x_i$ being $1,$ regarded as a module
over the mod-$2$ Steenrod algebra $\mathcal{A},$ and let $GL_k$ be the general
linear group over the prime field $\mathbb{F}_2$ which acts naturally on $P_k$.
We study the hit problem, set up by Frank Peterson, of finding a minimal set of
generators for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod
algebra, $\mathcal{A}$. These results are used to study the Singer algebraic
transfer which is a homomorphism from the homology of the mod-$2$ Steenrod
algebra, $\mbox{Tor}^{\mathcal{A}}_{k, k+n}(\mathbb{F}_2, \mathbb{F}_2),$ to
the subspace of $\mathbb{F}_2\otimes_{\mathcal{A}}P_k$ consisting of all the
$GL_k$-invariant classes of degree $n.$ In this paper, we explicitly compute
the hit problem for $k = 5$ and the degree $7.2^s-5$ with $s$ an arbitrary
positive integer. Using this result, we show that Singer's conjecture for the
algebraic transfer is true in the case $k=5$ and the above degree.
