Random Quantum LDPC Codes Approaching the Gilbert-Varshamov Bound

We show that for any $R,ε>0$ and any prime $p\geq 2$, there exists an infinite family of $p$-ary quantum low-density parity-check (QLDPC) codes, rate $R$, checks of weight $O_ε(1)$, and normalized distance at least $δ_{\mathrm{GV}}(p,R)-ε$. Here, $δ_{\mathrm{GV}}(p,R)$ denotes the quantum Gilbert-Va