TY - GEN

T1 - Combinatorial list-decoding of Reed-Solomon codes beyond the Johnson radius

AU - Shangguan, Chong

AU - Tamo, Itzhak

N1 - Publisher Copyright:
© 2020 ACM.

PY - 2020/6/8

Y1 - 2020/6/8

N2 - List-decoding of Reed-Solomon (RS) codes beyond the so called Johnson radius has been one of the main open questions in coding theory and theoretical computer science since the work of Guruswami and Sudan. It is now known by the work of Rudra and Wootters, using techniques from high dimensional probability, that over large enough alphabets there exist RS codes that are indeed list-decodable beyond this radius. In this paper we take a more combinatorial approach that allows us to determine the precise relation (up to the exact constant) between the decoding radius and the list size. We prove a generalized Singleton bound for a given list size, and conjecture that the bound is tight for most 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. Lastly, we give the first explicit construction 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 the notion of cycle space from Graph Theory. Both of them have not been used before in the context of list-decoding.

AB - List-decoding of Reed-Solomon (RS) codes beyond the so called Johnson radius has been one of the main open questions in coding theory and theoretical computer science since the work of Guruswami and Sudan. It is now known by the work of Rudra and Wootters, using techniques from high dimensional probability, that over large enough alphabets there exist RS codes that are indeed list-decodable beyond this radius. In this paper we take a more combinatorial approach that allows us to determine the precise relation (up to the exact constant) between the decoding radius and the list size. We prove a generalized Singleton bound for a given list size, and conjecture that the bound is tight for most 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. Lastly, we give the first explicit construction 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 the notion of cycle space from Graph Theory. Both of them have not been used before in the context of list-decoding.

KW - Generalized Singleton bound

KW - Johnson radius

KW - List-decoding

KW - Reed-Solomon codes

UR - http://www.scopus.com/inward/record.url?scp=85086759853&partnerID=8YFLogxK

U2 - 10.1145/3357713.3384295

DO - 10.1145/3357713.3384295

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AN - SCOPUS:85086759853

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 538

EP - 551

BT - STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

A2 - Makarychev, Konstantin

A2 - Makarychev, Yury

A2 - Tulsiani, Madhur

A2 - Kamath, Gautam

A2 - Chuzhoy, Julia

PB - Association for Computing Machinery

Y2 - 22 June 2020 through 26 June 2020

ER -