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 language | English |
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Pages (from-to) | 684-717 |
Number of pages | 34 |
Journal | SIAM Journal on Computing |
Volume | 52 |
Issue number | 3 |
DOIs | |
State | Published - Jun 2023 |
Keywords
- Johnson radius
- Reed-Solomon codes
- generalized Singleton bound
- list-decoding