Capacity-Achieving Codes with Inverse-Ackermann-Depth Encoders

We prove that for any additive noise channel over $\mathbb{F}_q$, there exist error-correcting codes approaching channel capacity encodable by arithmetic circuits (with weighted addition gates) over $\mathbb{F}_q$ of size $O(n)$ and depth $2\alpha(n)$, where $\alpha(n)$ is a version of the inverse Ackermann function that is at most $3$ for all input lengths $n$ in practice. Our results demonstrate that certain capacity-achieving codes admit highly efficient encoding circuits that are simultaneously of linear size and inverse-Ackermann depth. Our construction composes a linear code with constant rate and relative distance, based on the constructions of G\'{a}l, Hansen, Kouck\'{y}, Pudl\'{a}k, and Viola [IEEE Trans. Inform. Theory 59(10), 2013] and Drucker and Li [COCOON 2023], with an additional layer formed by a disperser graph. A probabilistic argument over the edge weights of the disperser shows the existence of a deterministic encoder achieving error probability $2^{-\Omega(n)}$ at any rate below capacity.