An improved bound for the length of matrix algebras

Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show that the $\mathbb{F}$-linear span of the words in $S$ of length at most $$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. This improves on the $n^2/3+2/3$ bound by Paz (1984) and an $O\left(n^{1.5}\right)$ bound of Pappacena (1997).