2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) 2018
DOI: 10.1109/focs.2018.00029
|View full text |Cite
|
Sign up to set email alerts
|

Improved Decoding of Folded Reed-Solomon and Multiplicity Codes

Abstract: In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions of codes known to achieve list-decoding capacity; multivariate multiplicity codes were the first constructions of high-rate locally correctable codes; and univariate multiplicity codes a… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1
1

Citation Types

0
7
0

Year Published

2019
2019
2023
2023

Publication Types

Select...
3
3
1

Relationship

1
6

Authors

Journals

citations
Cited by 24 publications
(9 citation statements)
references
References 32 publications
0
7
0
Order By: Relevance
“…For |Σ| = 2, the list decoding capacity is ρ = 1 − H 2 (α) − γ, while for |Σ| = ω (1), the list decoding capacity is ρ = 1 − α − γ. Over large alphabets, this tradeoff can be achieved by explicit codes with efficient list decoding algorithms [21] (see also [27] for the state of the art). Over binary alphabet, we do not know how to explicitly construct codes achieving list decoding capacity.…”
Section: List Decodingmentioning
confidence: 99%
“…For |Σ| = 2, the list decoding capacity is ρ = 1 − H 2 (α) − γ, while for |Σ| = ω (1), the list decoding capacity is ρ = 1 − α − γ. Over large alphabets, this tradeoff can be achieved by explicit codes with efficient list decoding algorithms [21] (see also [27] for the state of the art). Over binary alphabet, we do not know how to explicitly construct codes achieving list decoding capacity.…”
Section: List Decodingmentioning
confidence: 99%
“…Sian et al [15] proposed using RS codes as an iterative structure and presented an algebraic notation for the input seeds. According to Swastik et al [16], the RS codes can be defined using two fields: GF(q) and GF(q m ); however, in this paper, GF(q m ) was chosen to give a better transmission. For example, the RS codes can transmit 128 random bits for a seven-bit input message sequence in the finite field 'F'.…”
Section: Related Work Of the Lfsrmentioning
confidence: 99%
“…The maximum length pseudo-random patterns with the minimum distances are examined using the concepts of multiplicity in the codes. The multiplicity structure in the codes should be identified using the multivariant low-degree polynomial and the restriction of the univariant low-degree polynomial [16]. Therefore, the polynomial for the LFSR is fixed as low-degree multivariant bits.…”
Section: Minimum Distances Achieved For the Maximum Length Pseudo-random Test Patternsmentioning
confidence: 99%
“…Furthermore, if we use a folded Reed-Solomon code, then we still get a Shamir-like threshold secret sharing scheme, but in addition we can employ list-decoding when we are in a regime were decoding is not unique anymore. In summary, we have the following (see Appendix A.1 and [12] for the details).…”
Section: Shamir Secret Sharing With List Decodingmentioning
confidence: 99%
“…There are many works aimed at reducing the list size of the folded Reed-Solomon codes. Recently, Kopparty et al [12] proved that the list size of the folded Reed-Solomon codes is at most a constant in γ.…”
mentioning
confidence: 99%