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Let $n \ge d \ge \ell \ge 1$ be integers, and denote the $n$-dimensionalhypercube 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 ifevery $Q_d$ in the $Q_n$ contains a $Q_\ell$ of every color. In this paper weconsider 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 thatthere is a $d$-polychromatic linear $Q_\ell$-coloring of $Q_n$ for all $n \ged$. We prove that for all $d \ge 3$, $p_{lin}^{d-1}(d) = 2$. In addition, usinga 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.

Linear $d$-polychromatic $Q_{d-1}$-colorings of the Hypercube