Consider the product \begin{equation} P_{k,\ell} = \Pi_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{Z}_2[x_1,\ldots, x_k] \end{equation}

consisting of all products of all non-zero monomials $a_1x_1+\ldots +a_kx_k$, at most $\ell$ of whose coefficients are non-zero. 

Letting \begin{equation} \alpha_{j,\ell} = \sum_{i=0}^{\ell-1}\binom{j-1}{i}, \end{equation} 
it is not hard to confirm that $P_{k,\ell}$ can be expressed as the following determinant: 

\begin{equation} P_{k,\ell}= Det \begin{pmatrix} x_1^{\alpha_{1,\ell}} & \cdots & x_k^{\alpha_{1,\ell}} \\
\vdots & \vdots & \vdots \\
x_1^{\alpha_{k,\ell}} & \cdots & x_k^{\alpha_{k,\ell}} \end{pmatrix} = \sum_{\sigma \in S_k} x_{\sigma(1)}^{\alpha_{k,\ell}}x_{\sigma(2)}^{\alpha_{k-1,\ell}}\cdots x_{\sigma(k)}^{\alpha_{1,\ell}},  \end{equation} 

$S_k$ the symmetric group.

(Note that $\alpha_{1,\ell}=1$, and we assume as you usual that $\binom{a}{b}=0$ if $a<b$.) 

**Question**: I am sure the formula must be classically known (e.g., it is a Dickson polynomial when $\ell=k$). I would be grateful for a reference.