  Let $n \ge d \ge \ell \ge 1$ be integers, and denote the $n$-dimensional
hypercube by $Q_n$. A coloring of the $\ell$-dimensional subcubes $Q_\ell$ in
$Q_n$ is called a $Q_\ell$-coloring. Such a coloring is $d$-polychromatic if
every $Q_d$ in the $Q_n$ contains a $Q_\ell$ of every color. In this paper we
consider a specific class of $Q_\ell$-colorings that are called linear. Given
$\ell$ and $d$, let $p_{lin}^\ell(d)$ be the largest number of colors such that
there is a $d$-polychromatic linear $Q_\ell$-coloring of $Q_n$ for all $n \ge
d$. We prove that for all $d \ge 3$, $p_{lin}^{d-1}(d) = 2$. In addition, using
a computer search, we determine $p_{lin}^\ell(d)$ for some specific values of
$\ell$ and $d$, in some cases improving on previously known lower bounds.
