Linear-Time Erasure List-Decoding of Expander Codes

Ron-Zewi, Noga; Wootters, Mary; Zémor, Gilles

doi:10.1109/tit.2021.3086805

Cited by 4 publications

(1 citation statement)

References 34 publications

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“…There exist highly efficient algorithms for the unique-decoding of these codes from both probabilistic and adversarial errors, based on combinatorial arguments, linear programming relaxations [Fel03,ADS12,FWK05] and message passing algorithms [Gur06,RU08]. In the setting of erasures where the location of the corruptions in the transmitted codeword is known, recent work has also led to linear-time list decoding algorithms [RZWZ21,HW18], which also work for the more general task of list recovery in the large alphabet (high-rate) case [HW18]. However, to the best of our knowledge, no list decoding algorithms are known in the more challenging (and common) setting of errors when the location of the corruptions are unknown, even though random ensembles of LDPC codes are even known to combinatorially achieve list-decoding capacity [MRRZ + 20], and thus have bounded list sizes up to optimal error radii.…”

Section: Tanner Codesmentioning

confidence: 99%

Near-linear time decoding of Ta-Shma’s codes via splittable regularity

Jeronimo

Srivastava

Tulsiani

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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We develop new list decoding algorithms for Tanner codes and distance-amplified codes based on bipartite spectral expanders. We show that proofs exhibiting lower bounds on the minimum distance of these codes can be used as certificates discoverable by relaxations in the Sum-of-Squares (SoS) semidefinite programming hierarchy. Combining these certificates with certain entropic proxies to ensure that the solutions to the relaxations cover the entire list, then leads to algorithms for list decoding several families of codes up to the Johnson bound. We prove the following results:-We show that the LDPC Tanner codes of Sipser-Spielman [IEEE Trans. Inf. Theory 1996] andZémor [IEEE Trans. Inf. Theory 2001] with alphabet size q, block-length n and distance δ, based on an expander graph with degree d, can be list-decoded up to distance J q (δ) − ε in time n O d,q (1/ε 4 ) , where J q (δ) denotes the Johnson bound.-We show that the codes obtained via the expander-based distance amplification procedure of Alon, Edmonds and Luby [FOCS 1995] can be list-decoded close to the Johnson bound using the SoS hierarchy, by reducing the list decoding problem to unique decoding of the base code.In particular, starting from any base code unique-decodable up to distance δ, one can obtain near-MDS codes with rate R and distance 1 − R − ε, list-decodable up to the Johnson bound in time n O ε,δ (1) .-We show that the locally testable codes of Dinur et al. [STOC 2022] with alphabet size q, block-length n and distance δ based on a square Cayley complex with generator sets of size d, can be list-decoded up to distance J q (δ) − ε in time n O d,q (1/ε 4 ) , where J q (δ) denotes the Johnson bound.

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Section: Tanner Codesmentioning

confidence: 99%

Near-linear time decoding of Ta-Shma’s codes via splittable regularity

Jeronimo

Srivastava

Tulsiani

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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List Decoding of Tanner and Expander Amplified Codes from Distance Certificates

Jeronimo,

Srivastava,

Tulsiani

2023

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)

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Improved Decoding of Expander Codes

Chen

Cheng

et al. 2023

IEEE Trans. Inform. Theory

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We study the classical expander codes, introduced by Sipser and Spielman [10]. Given any constants 0 < α, ε < 1/2, and an arbitrary bipartite graph with N vertices on the left, M < N vertices on the right, and left degree D such that any left subset S of size at most αN has at least (1 − ε)|S|D neighbors, we show that the corresponding linear code given by parity checks on the right has distance at least roughly αN 2ε . This is strictly better than the best known previous result of 2(1 − ε)αN [11, 12] whenever ε < 1/2, and improves the previous result significantly when ε is small. Furthermore, we show that this distance is tight in general, thus providing a complete characterization of the distance of general expander codes.Next, we provide several efficient decoding algorithms, which vastly improve previous results in terms of the fraction of errors corrected, whenever ε < 1 4 . Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has a high rate).Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests.

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Linear-Time Erasure List-Decoding of Expander Codes

Cited by 4 publications

References 34 publications

Near-linear time decoding of Ta-Shma’s codes via splittable regularity

Near-linear time decoding of Ta-Shma’s codes via splittable regularity

List Decoding of Tanner and Expander Amplified Codes from Distance Certificates

Improved Decoding of Expander Codes

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