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 $2α(n)$, where $α(n)$ 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 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. A probabilistic argument over the edge weights of the disperser shows the existence of a deterministic encoder achieving error probability $2^{-Ω(n)}$ at any rate below capacity.

翻译：我们证明，对于任意加法噪声信道（加性噪声通道），存在逼近信道容量的纠错码，这些纠错码可由大小为$O(n)$、深度为$2α(n)$的算术电路（含加权加法门）在$\mathbb{F}_q$上编码，其中$α(n)$是逆阿克曼函数的一个变体，当输入长度$n$在实际范围内时，该函数值至多为3。我们的结果表明，某些达到容量的码同时具有线性规模和逆阿克曼深度的极高效率编码电路。我们的构造基于Gál、Hansen、Koucký、Pudlák和Viola [IEEE Trans. Inform. Theory 59(10), 2013] 以及Drucker和Li [COCOON 2023]的构造，将具有恒定码率和相对距离的线性码与由分散器图形成的附加层相结合，构成复合码。对分散器边权的概率论证表明，存在一种确定性编码器，在容量以下的任何码率下都能实现$2^{-Ω(n)}$的错误概率。