Average-radius list-recoverability of random linear codes

Rudra, Atri; Wootters, Mary

doi:10.1137/1.9781611975031.42

Cited by 16 publications

(22 citation statements)

References 21 publications

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“…On the downside, we note that our arguments only work for binary codes and do not extend to larger alphabets; additionally, our positive results do not establish average-radius list-decodability with list size O(1/ε), a stronger notion which was established in some of the previous works [2,43,36]. It would be very interesting to extend our results to these settings.…”

Section: :3mentioning

confidence: 61%

“…A unified analysis. As we discuss more below, previous work on the list-decodability of random linear binary codes either work only in certain (non-overlapping) parameter regimes [12,43], or else get substantially sub-optimal bounds on the list-size [36]. Our argument obtains improved list size bounds over all these results and works in all parameter regimes.…”

Section: :3mentioning

confidence: 76%

“…Because it is a major open problem to construct explicit binary codes of rate 1 − H(p) − ε with constant (or even poly(n)) list-sizes, one natural line of work has been to study structured random approaches, in particular random linear codes. A random linear code C ⊂ F n 2 is simply a random subspace of F n 2 , and the list-decodability of these codes has been wellstudied [45,13,12,2,43,34,36]. There are several reasons to study the list-decodability of random linear codes.…”

Section: Introductionmentioning

confidence: 99%

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Improved List-Decodability of Random Linear Binary Codes

Wootters

2021

IEEE Trans. Inform. Theory

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There has been a great deal of work establishing that random linear codes are as list-decodable as uniformly random codes, in the sense that a random linear binary code of rate 1 − H(p) − is (p, O(1/))-list-decodable with high probability. In this work, we show that such codes are (p, H(p)/ + 2)-list-decodable with high probability, for any p ∈ (0, 1/2) and > 0. In addition to improving the constant in known list-size bounds, our argument-which is quite simple-works simultaneously for all values of p, while previous works obtaining L = O(1/) patched together different arguments to cover different parameter regimes. Our approach is to strengthen an existential argument of (Guruswami, Håstad, Sudan and Zuckerman, IEEE Trans. IT, 2002) to hold with high probability. To complement our upper bound for random linear binary codes, we also improve an argument of (Guruswami, Narayanan, IEEE Trans. IT, 2014) to obtain a tight lower bound of 1/ on the list size of uniformly random binary codes; this implies that random linear binary codes are in fact more list-decodable than uniformly random binary codes, in the sense that the list sizes are strictly smaller. To demonstrate the applicability of these techniques, we use them to (a) obtain more information about the distribution of list sizes of random linear binary codes and (b) to prove a similar result for random linear rank-metric codes.

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Section: :3mentioning

confidence: 61%

Section: :3mentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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Improved List-Decodability of Random Linear Binary Codes

Wootters

2021

IEEE Trans. Inform. Theory

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“…We use the following result on the list-recoverability of random linear codes from [31]. Proof of Corollary 7.…”

Section: Corollarymentioning

confidence: 99%

On List Recovery of High-Rate Tensor Codes

Kopparty

Resch

Ron-Zewi

et al. 2021

IEEE Trans. Inform. Theory

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We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is approximately locally list recoverable, as well as globally list recoverable in probabilistic near-linear time. This was used in turn to give the first capacity-achieving list decodable codes with (1) local list decoding algorithms, and with (2) probabilistic near-linear time global list decoding algorithms. This also yielded constant-rate codes approaching the Gilbert-Varshamov bound with probabilistic near-linear time global unique decoding algorithms. In the current work we obtain the following results: 1. The tensor product of an efficient (poly-time) high-rate globally list recoverable code is globally list recoverable in deterministic near-linear time. This yields in turn the first capacity-achieving list decodable codes with deterministic near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert-Varshamov bound with deterministic near-linear time global unique decoding algorithms. 2. If the base code is additionally locally correctable, then the tensor product is (genuinely) locally list recoverable. This yields in turn (non-explicit) constant-rate codes approaching the Gilbert-Varshamov bound that are locally correctable with query complexity and running time N o(1). This improves over prior work by Gopi et. al. (SODA'17; IEEE Transactions on Information Theory'18) that only gave query complexity N ε with rate that is exponentially small in 1/ε. 3. A nearly-tight combinatorial lower bound on output list size for list recovering high-rate tensor codes. This bound implies in turn a nearly-tight lower bound of N Ω(1/ log log N) on the product of query complexity and output list size for locally list recovering high-rate tensor codes. 2012 ACM Subject Classification Mathematics of computing → Coding theory; Theory of computation → Pseudorandomness and derandomization Keywords and phrases Coding theory, Tensor codes, List-decoding and recovery, Local codes

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“…However, it is easier to analyze since the problem is linearized from infinity norm to one norm. Indeed it plays a useful role in a long line of work towards understanding the list decodability of random linear codes[26,45,36,37,38].…”

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confidence: 99%

Generalized List Decoding

Zhang

Budkuley

Jaggi

2020

2020 Information Theory and Applications Workshop (ITA)

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This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip (XOR) channels, erasure channels, AND (Z-) channels, OR (Z-) channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) pL´1q-list decodable codes (where the list size L is a universal constant) exist for such channels. Our criterion essentially asserts that: 2012 ACM Subject Classification Mathematics of computing Ñ Coding theory; Mathematics of computing Ñ Information theory Keywords and phrases Generalized Plotkin bound, general adversarial channels, equicoupled codes, random coding, completely positive tensors, copositive tensors, hypergraph Ramsey theory

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Average-radius list-recoverability of random linear codes

Cited by 16 publications

References 21 publications

Improved List-Decodability of Random Linear Binary Codes

Improved List-Decodability of Random Linear Binary Codes

On List Recovery of High-Rate Tensor Codes

Generalized List Decoding

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