Capacity-Achieving Codes with Inverse-Ackermann-Depth Encoders
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911602354159616 |
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| author | Li, Yuan |
| author_facet | Li, Yuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11443 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Capacity-Achieving Codes with Inverse-Ackermann-Depth Encoders Li, Yuan Information Theory 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. |
| title | Capacity-Achieving Codes with Inverse-Ackermann-Depth Encoders |
| topic | Information Theory |
| url | https://arxiv.org/abs/2512.11443 |