Capacity-Achieving Codes with Inverse-Ackermann-Depth Encoders

For any symmetric discrete memoryless channel with input and output alphabet of size $q$, where $q$ is a prime power, we prove that 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 $α(n)$, where $α(n)$ is a version of the inverse Ackermann function. Our results suggest that certain capacity-achieving codes admit highly efficient encoding circuits that are both in linear size and of inverse-Ackermann depth. Our construction composes a linear code with constant rate and relative distance, based on the constructions of Gál, Hansen, Koucký, Pudlák, and Viola [IEEE Trans. Inform. Theory 59(10), 2013] and Drucker and Li [COCOON 2023], with an additional layer formed by a disperser graph whose edge weights are chosen uniformly at random.