2024/10/20 by Xin Li, Li, Xin, Mao, Songtao · 1 citation
Computer Science · Engineering · #Combinatorics (math.CO) #Data Structures and Algorithms (cs.DS) #Error Correcting Code Techniques #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Radar Systems and Signal Processing #Wireless Communication Security Techniques
paper · pdf · doi:10.48550/arxiv.2410.15506
openalex publication_date 2024/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from (1-ε)/(2) fraction of errors and list decodable from 1-ε 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 \mathbbF2-linear codes with rate Ω(ε) and alphabet size 2poly log(1/ε), that are capable of correcting e errors and s erasures whenever 2e + s < (1 - ε)n in linear-time. 2. Explicit Near-Optimal List Decodable Codes: We construct a family of explicit list decodable codes with rate Ω(ε) and alphabet size 2poly log(1/ε), that are capable of list decoding from 1-ε fraction of errors with a list size L = expexpexp(log∗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/ε), albeit with a suboptimal rate of O(ε2), capable of list decoding from 1-ε 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 (ε)/(log2(1/ε)) and list size (log2(1/ε))/(ε), 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.