Threshold Rates for Properties of Random Codes

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

doi:10.1109/tit.2021.3123497

Cited by 3 publications

(14 citation statements)

References 25 publications

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“…In [15], a lower bound for the list-decodability of capacity-approaching random linear codes is given, showing that lists of size ∼ hq(ρ) ε are required: our list-recovery list-size lower bound is a generalization of this result. Lastly, in [17] the threshold rate for (ρ, 2)list-decodability is computed, providing a lower bound and an upper bound: this segues us nicely into a discussion of the work on computing upper bounds on list-sizes.…”

Section: Related Workmentioning

confidence: 97%

“…This toolkit was developed by Mosheiff et al [28] on the way to proving that LDPC codes achieve list-decoding capacity; recent works [15,16] have found further uses for the techniques in investing combinatorial properties of random linear codes. An analogous threshold toolkit for random codes was provided in [17].…”

Section: Techniquesmentioning

confidence: 99%

“…Broadly speaking, the techniques of [28,17] apply when considering a property of codes defined by forbidding a family of "bad" subsets, each of which have constant cardinality (independent of n). For example, the property of (ρ, L)-list-decodability is defined by forbidding all L-element subsets B = {x 1 , .…”

Section: Techniquesmentioning

confidence: 99%

“…Random Codes. For the case of random codes, we can compute the threshold rates for all the properties of interest in a relatively straightforward way, as the characterization from [17] does not require us to consider any sort of compressing mapping on the types. Quite notably, in all cases we see that random linear codes appear to perform better than random codes.…”

Section: :8 Threshold Rates Of Code Ensembles: Linear Is Bestmentioning

confidence: 99%

“…In Section 1.2 we outlined the works [28,15,17] which developed and studied the thresholds toolkit that we apply. In this section, we provide more context for the study of random linear codes and their list-decodability/-recoverability.…”

Section: Related Workmentioning

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: Related Workmentioning

confidence: 97%

Section: Techniquesmentioning

confidence: 99%

Section: Techniquesmentioning

confidence: 99%

Section: :8 Threshold Rates Of Code Ensembles: Linear Is Bestmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 3 more Smart Citations

Threshold Rates of Code Ensembles: Linear Is Best

Resch,

Yuan

2024

IEEE Trans. Inform. Theory

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

Guruswami

Mosheiff

et al. 2022

IEEE Trans. Inform. Theory

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Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

Resch,

Yuan,

Zhang

2024

IEEE Trans. Inform. Theory

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In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of q ě 2. A code is called pp, Lqq-list-decodable if every radius pn Hamming ball contains less than L codewords; pp, ℓ, Lqq-list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length ℓ and again stipulate that there be less than L codewords.Our main contribution is to precisely calculate the maximum value of p for which there exist infinite families of positive rate pp, ℓ, Lqq-list-recoverable codes, the quantity we call the zero-rate threshold. Denoting this value by p˚, we in fact show that codes correcting a p˚`ε fraction of errors must have size Oεp1q, i.e., independent of n. Such a result is typically referred to as a "Plotkin bound." To complement this, a standard random code with expurgation construction shows that there exist positive rate codes correcting a p˚´ε fraction of errors. We also follow a classical proof template (typically attributed to Elias and Bassalygo) to derive from the zero-rate threshold other tradeoffs between rate and decoding radius for list-decoding and list-recovery.Technically, proving the Plotkin bound boils down to demonstrating the Schur convexity of a certain function defined on the q-simplex as well as the convexity of a univariate function derived from it. We remark that an earlier argument claimed similar results for q-ary list-decoding; however, we point out that this earlier proof is flawed.

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Threshold Rates for Properties of Random Codes

Cited by 3 publications

References 25 publications

Threshold Rates of Code Ensembles: Linear Is Best

Threshold Rates of Code Ensembles: Linear Is Best

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

Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

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