Capacity-Achieving Codes with Inverse-Ackermann-Depth Encoders

TL;DR

This paper constructs capacity-approaching error-correcting codes that can be encoded by highly efficient arithmetic circuits with near-constant inverse-Ackermann depth, combining advanced coding and graph techniques.

Contribution

It introduces a new class of capacity-achieving codes with encoders of linear size and extremely shallow inverse-Ackermann depth, using a novel composition of linear codes and disperser graphs.

Findings

Codes approach channel capacity with error probability exponentially small in n.

Encoders have size O(n) and depth 2α(n), with α(n) at most 3 in practice.

Construction combines existing codes with a disperser graph layer.

Abstract

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.