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

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