Improved Explicit Near-Optimal Codes in the High-Noise Regimes

TL;DR

This paper presents new explicit code constructions that are near-optimal in rate and decoding efficiency for high-noise regimes, including unique and list decoding with improved parameters and algorithms.

Contribution

It introduces several explicit, near-optimal codes with efficient decoding algorithms for high-noise error correction, including novel combinatorial objects and decoding methods.

Findings

Constructed linear-time uniquely decodable codes with high rate.

Developed list decodable codes with extremely large list sizes in polynomial time.

Achieved near-optimal list size with polynomial-time decoding using new combinatorial objects.

Abstract

We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of errors. We present several improved explicit constructions that achieve near-optimal rates, as well as efficient or even linear-time decoding algorithms. Our contributions are as follows. 1. Explicit Near-Optimal Linear Time Uniquely Decodable Codes: We construct a family of explicit $\mathbb{F}_2$-linear codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of correcting $e$ errors and $s$ erasures whenever $2e + s < (1 - \varepsilon)n$ in linear-time. 2. Explicit Near-Optimal List Decodable Codes: We construct a family of explicit list decodable codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of list decoding from $1-\varepsilon$ fraction of errors with a list size $L = \exp\exp\exp(\log^{\ast}n)$ in polynomial time. 3. List Decodable Code with Near-Optimal List Size: We construct a family of explicit list decodable codes with an optimal list size of $O(1/\varepsilon)$, albeit with a suboptimal rate of $O(\varepsilon^2)$, capable of list decoding from $1-\varepsilon$ fraction of errors in polynomial time. Furthermore, we introduce a new combinatorial object called multi-set disperser, and use it to give a family of list decodable codes with near-optimal rate $\frac{\varepsilon}{\log^2(1/\varepsilon)}$ and list size $\frac{\log^2(1/\varepsilon)}{\varepsilon}$, that can be constructed in probabilistic polynomial time and decoded in deterministic polynomial time. We also introduce new decoding algorithms that may prove valuable for other graph-based codes.