Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity

We prove the existence of Reed-Solomon codes of any desired rate R ∈ (0, 1) that are combinatorially list-decodable up to a radius approaching 1 − R, which is the informationtheoretic limit. This is established by starting with the full-length [q, k]q Reed-Solomon code over a field Fq that is polynomially larger than the desired dimension k, and “puncturing” it by including k/R randomly chosen codeword positions. Our puncturing result is more general and applies to any code with large minimum distance: we show that a random rate R puncturing of an Fq-linear “mother” code whose relative distance is close enough to 1 − 1/q is list-decodable up to a radius approaching the q-ary list-decoding capacity bound h−1 q (1 − R). In fact, for large q, or under a stronger assumption of low-bias of the mother-code, we prove that the threshold rate for list-decodability with a specific list-size (and more generally, any “local” property) of the random puncturing approaches that of fully random linear codes. Thus, all current (and future) list-decodability bounds shown for random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) code. This can be viewed as a general derandomization result applicable to random linear codes. To obtain our conclusion about Reed-Solomon codes, we establish some hashing properties of field trace maps that allow us to reduce the list-decodability of RS codes to its associated trace (dual-BCH) code, and then apply our puncturing theorem to the latter. Our approach implies, essentially for free, optimal rate list-recoverability of punctured RS codes as well. Research supported in part by NSF grants CCF-1814603 and CCF-1908125, and a Simons Investigator Award. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. ar X iv :2 10 9. 11 72 5v 2 [ cs .C C ] 8 N ov 2 02 1