Bounds for List-Decoding and List-Recovery of Random Linear Codes

Guruswami, Venkatesan; Li, Ray; Mosheiff, Jonathan; Resch, Nicolas; Silas, Shashwat; Wootters, Mary

doi:10.1109/tit.2021.3127126

Cited by 9 publications

(16 citation statements)

References 39 publications

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“…▶ Remark 3. It might appear that our conjecture that random linear codes outperform random codes for list-recovery is contradicted by the result of [15]. However, we emphasize that the capacity for erasure list-recovery is larger, so if a code is ε-close to capacity for list-recovery from erasures for small ε > 0 it is above capacity for list-recovery from errors, the model we study.…”

Section: Sourcementioning

confidence: 70%

“…For sufficiently small ε > 0, This table summarizes much of the work on the list-recoverability of random linear codes (RLC) and random codes (RC). The lower bound of [15] only applies when q = p Ω(1/ε) for a prime p, and in [32] η > 0 is viewed as a small constant. [15] also offers a similar lower bound for the case of list-recovery from erasures.…”

Section: Our Resultsmentioning

confidence: 99%

“…The lower bound of [15] only applies when q = p Ω(1/ε) for a prime p, and in [32] η > 0 is viewed as a small constant. [15] also offers a similar lower bound for the case of list-recovery from erasures.…”

Section: Our Resultsmentioning

confidence: 99%

“…We therefore conjecture that Theorem 1 is indeed tight. This stands in stark contrast to erasure list-recovery: 3 for this model, it is known that random linear codes can require lists of size ℓ Ω(1/ε) [15] (at least, if the field has large characteristic), whereas the lists for random codes can be just O(ℓ/ε). A summary of the state-of-the-art for list-recovery of RLCs and RCs is provided in Table 1.…”

Section: Sourcementioning

confidence: 99%

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Threshold Rates of Code Ensembles: Linear Is Best

Resch,

Yuan

2024

IEEE Trans. Inform. Theory

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In this work, we prove new results concerning the combinatorial properties of random linear codes. By applying the thresholds framework from Mosheiff et al. (FOCS 2020) we derive fine-grained results concerning the list-decodability and -recoverability of random linear codes.Firstly, we prove a lower bound on the list-size required for random linear codes over Fq ε-close to capacity to list-recover with error radius ρ and input lists of size ℓ. We show that the list-size, where R is the rate of the random linear code. This is analogous to a lower bound for list-decoding that was recently obtained by Guruswami et al. (IEEE TIT 2021B). As a comparison, we also pin down the list size of random codes which is log q ( q ℓ ) ε . This result almost closes the O( q log L L ) gap left by Guruswami et al. (IEEE TIT 2021A). This leaves open the possibility (that we consider likely) that random linear codes perform better than the random codes for list-recoverability, which is in contrast to a recent gap shown for the case of list-recovery from erasures (Guruswami et al., IEEE TIT 2021B).Next, we consider list-decoding with constant list-sizes. Specifically, we obtain new lower bounds on the rate required for: List-of-3 decodability of random linear codes over F2;List-of-2 decodability of random linear codes over Fq (for any q). This expands upon Guruswami et al. (IEEE TIT 2021A) which only studied list-of-2 decodability of random linear codes over F2. Further, in both cases we are able to show that the rate is larger than that which is possible for uniformly random codes.A conclusion that we draw from our work is that, for many combinatorial properties of interest, random linear codes actually perform better than uniformly random codes, in contrast to the apparently standard intuition that uniformly random codes are best.

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Section: Sourcementioning

confidence: 70%

Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Sourcementioning

confidence: 99%

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Threshold Rates of Code Ensembles: Linear Is Best

Resch,

Yuan

2024

IEEE Trans. Inform. Theory

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“…Though there have been active research on list-decodable codes since 1997, few general bounds on list-decodable codes have been obtained. We refer to early papers [59,32,33,37] for the asymptotical combinatorial bounds of list decodable codes and the recent papers [82,46]. The classical Singleton bound |C| ≤ q n−2d list for the (d list , 1) list-decodable codes in [84] was generalized to…”

Section: Related Workmentioning

confidence: 99%

List-decodable Codes and Covering Codes

Chen¹

2021

Preprint

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The list-decodable code has been an active topic in theoretical computer science since the seminal papers of M. Sudan and V. Guruswami in 1997-1998. There are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. However few results about general constraints on rates, list-decodable radius and list sizes for list-decodable codes have been obtained. List-decodable codes are also considered in rank-metric, subspace metric, cover-metric, pair metric and insdel metric settings. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes. The general covering code upper bounds can be applied to the case that the volumes of the balls depend on the centers, not only on the radius. Then any good upper bound on the covering radius or the size of covering code imply a good upper bound on the sizes of list-decodable codes. Hence the list-decodablity of codes is a strong constraint from the view of covering codes on general finite metric spaces. Our results give exponential improvements on the recent generalized Singleton upper bound of Shangguan and Tamo in STOC 2020 for Hamming metric list-decodable codes, when the code lengths are large. The generalized Singleton upper bound on the average-radius list-decodable codes is also given from our general covering code upper bound. The asymptotic forms of covering code bounds can partially recover the Blinovsky bound and the combinatorial bound of Guruswami-Håstad-Sudan-Zuckerman in Hamming metric setting. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial listdecodable codes. We apply our general covering code upper bounds for

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