GENERALIZED SINGLETON BOUND AND LIST-DECODING REED-SOLOMON CODES BEYOND THE JOHNSON RADIUS

Chong Shangguan, Itzhak Tamo

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we take a combinatorial approach to the problem of list-decoding, which allows us to determine the precise relation (up to the exact constant) between the decoding radius, list size, and code rate. We prove a generalized Singleton bound for a given list size, and conjecture that the bound is tight for most Reed-Solomon (RS) codes over large enough finite fields. We also show that the conjecture holds true for list sizes 2 and 3, and as a by product show that most RS codes with a rate of at least 1/9 are list-decodable beyond the Johnson radius. Last, we give the first explicit construction in the literature of such RS codes. The main tools used in the proof are a new type of linear dependency between codewords of a code that are contained in a small Hamming ball, and a surprising connection between list-decoding and the notion of cycle space in graph theory. Both of them are new, and may be of independent interest.

Original languageEnglish
Pages (from-to)684-717
Number of pages34
JournalSIAM Journal on Computing
Volume52
Issue number3
DOIs
StatePublished - Jun 2023

Funding

FundersFunder number
NSF-BSF2015814
European Research Council852953
National Natural Science Foundation of China12101364
Israel Science Foundation1030/15
Tel Aviv University
Natural Science Foundation of Shandong ProvinceZR2021QA005
National Key Research and Development Program of China2021YFA1001000

    Keywords

    • Johnson radius
    • Reed-Solomon codes
    • generalized Singleton bound
    • list-decoding

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