Counting Cholesky factorizations of the zero matrix over ${\mathbb{F}}_2$

A square, upper-triangular matrix $U$ is a Cholesky root of a matrix $M$ provided $U^{\ast }U=M$, where ${\cdot }^{\ast }$ represents the conjugate transpose when working over the complex field and $U^{\ast }=U^T$ over the reals and finite fields. In this paper, we investigate the number of such factorizations over the finite field with two elements, ${\mathbb{F}}_2$, and prove the equinumerosity, for each fixed rank, of the Cholesky roots of and the upper-triangular square roots of the zero matrix. We then provide asymptotics for this count and finish with a few directions for future inquiry.