On the list decodability of random linear codes with large error rates

Wootters, Mary

doi:10.1145/2488608.2488716

Cited by 29 publications

(56 citation statements)

References 24 publications

(29 reference statements)

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“…This is the setting studied in [CGV13,Woo13,RW14] and is relevant for related notions in pseudorandomness, like expanders, extractors, and hardness amplification [Vad11]. More precisely, for a large alphabet size q, we consider the problem of (1−ε, L)-list-decodability, or (ε, , L)-list-recovery, when ε = 1/q + δ, for some small δ and for q δ −1 .…”

Section: Results and Related Workmentioning

confidence: 99%

“…The list-decodability of random linear codes, and related notions, has been studied for decades in a variety of parameter regimes [ZP82,Eli91,GHK11,CGV13,Woo13,RW14].…”

Section: Introductionmentioning

confidence: 99%

“…Second and more surprisingly, it turns out that it is actually often more natural to prove upper bounds on average-radius list-decodability, as the sum is nicer to work with than the maximum. This was the approach taken in [CGV13,Woo13,RW14,RW15]. Further, most natural bounds (both upper and lower) for list-decoding apply to average-radius list-decoding, notably the listdecoding capacity theorem mentioned above.…”

Section: Introductionmentioning

confidence: 99%

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Average-radius list-recoverability of random linear codes

Rudra

Wootters

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We analyze the list-decodability, and related notions, of random linear codes. This has been studied extensively before: there are many different parameter regimes and many different variants. Previous works have used complementary styles of arguments-which each work in their own parameter regimes but not in othersand moreover have left some gaps in our understanding of the list-decodability of random linear codes. In particular, none of these arguments work well for listrecovery, a generalization of list-decoding that has been useful in a variety of settings.In this work, we present a new approach, which works across parameter regimes and further generalizes to list-recovery. In particular, our argument provides better results for list-decoding and list-recovery over large fields; improved (quasipolynomial) list sizees for high-rate list-recovery of random linear codes; improved algorithmic results for list-decoding; and optimal average-radius list-decoding over constant-sized alphabets.

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Section: Results and Related Workmentioning

confidence: 99%

“…The list-decodability of random linear codes, and related notions, has been studied for decades in a variety of parameter regimes [ZP82,Eli91,GHK11,CGV13,Woo13,RW14].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Average-radius list-recoverability of random linear codes

Rudra

Wootters

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The dependence of C q,ρ , however, degrades badly as the error fraction ρ approaches the maximum possible value of 1 − 1/q. Follow-up works [CGV13,Woo13,RW14] have addressed this issue, obtaining optimal bounds also in the high-error regime (using very different techniques).…”

Section: Prior Resultsmentioning

confidence: 99%

On the List-Decodability of Random Linear Rank-Metric Codes

Guruswami

Resch

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an F q -linear rank-metric code over F m×n q of rate R = (1 − ρ)(1 − n m ρ) − ε is shown to be (with high probability) list-decodable up to fractional radius ρ ∈ (0, 1) with lists of size at most Cρ,q ε , where C ρ,q is a constant depending only on ρ and q. This matches the bound for random rank-metric codes (up to constant factors). The proof adapts the approach of Guruswami, Håstad, Kopparty (STOC 2010), who established a similar result for the Hamming metric case, to the rank-metric setting. IntroductionAt its core, coding theory studies how many elements of a (finite) vector space one can pack subject to the constraint that no two elements are too close. Typically, the notion of closeness is that of Hamming distance, that is, the distance between two vectors is the number of coordinates on which they differ. In a rank-metric code, introduced in [Del78], codewords are matrices over a finite field and the distance between codewords is the rank of their difference. A linear rank-metric code is a subspace of matrices (over the field to which the matrix entries belong) such that every non-zero matrix in the subspace has large rank.Rank-metric codes have found applications in magnetic recording [Rot91], public-key cryptography [GPT91, Loi10, Loi17], and space-time coding [LGB03, LK05]. There has been a resurgence of interest in this topic due to the utility of rank-metric codes and the closely related subspace codes for error-control in random network coding [KK08, SKK08]. Decoding algorithms for rank-metric codes also have connections to the popular topic of low-rank recovery, specifically in a formulation where the task is to recover a matrix H from few inner products H, M with measurement matrices M [FS12]. Finally, the study of rank-metric codes raises additional mathematical and algorithmic

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“…q ě 2 p " p1´1{qqp1´τ q R " Ω´τ 2 log 3 pq{τ q log q¯L " Op1{τ 2 q with constant probability [13] q ě 2 p " p1´1{qqp1´τ q R " Ωpτ 2 { log qq L " Op1{τ 2 q whp [14] q " 2 p P p0, 1{2q R " 1´Hppq´δ L ď Hppq{δ`2 whp [10] random linear codes degenerate in the sense that the list size blows up as a function of δ Ñ 0. Another extreme case which is tricky to handle is when the field size q is very large and can be an increasing function of δ Ñ 0.…”

Section: Field Sizementioning

confidence: 99%

List Decoding Random Euclidean Codes and Infinite Constellations

Zhang

Vatedka

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints P and N respectively, the list decoding capacity is equal to 1 2 log P N . Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower bound on the list size of typical random spherical codes. We also give an upper bound on the list size achievable using nested Construction-A lattices and infinite Construction-A lattices. We then define and study a class of infinite constellations that generalize Construction-A lattices and prove upper and lower bounds for the same. Other goodness properties such as packing goodness and AWGN goodness of infinite constellations are proved along the way. Finally, we consider random lattices sampled from the Haar distribution and show that if a certain number-theoretic conjecture is true, then the list size grows as a polynomial function of the gap-to-capacity.

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On the list decodability of random linear codes with large error rates

Cited by 29 publications

References 24 publications

Average-radius list-recoverability of random linear codes

Average-radius list-recoverability of random linear codes

On the List-Decodability of Random Linear Rank-Metric Codes

List Decoding Random Euclidean Codes and Infinite Constellations

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