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List Decoding

Sushil Jajodia, Pierangela Samarati, Moti Yung · 2025

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List Decoding

Volodia Blinovsky · 1997

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List Decoding

2009

Cited by 3

List Decoding

Ian F. Blake · Cambridge University Press eBooks · 2024

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List Decoding

Rolando Antonio Carrasco BEng(Hons), PhD, Dr Martin Johnston BSc(Hons), MSc, PhD · 2008

This chapter contains sections titled: Introduction List Decoding of Reed–Solomon Codes Using the Guruswami–Sudan Algorithm Soft-Decision List Decoding of Reed-Solomon Codes Using the Kötter–Vardy Algorithm List Decoding of Algebraic–Geometric Codes Determining the Corresponding Coefficients Complexity Reduction Interpolation General Factorization Soft-Decision List Decoding of Hermitian Codes Conclusions References

Cited by 4

Generalized List Decoding

Yihan Zhang, Amitalok J. Budkuley, Sidharth Jaggi · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2020

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 channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) (L-1)-list decodable codes (where the list size L is a universal constant) exist for such channels. Our criterion essentially asserts that:
\nFor any given general adversarial channel, it is possible to construct positive rate (L-1)-list decodable codes if and only if the set of completely positive tensors of order-L with admissible marginals is not entirely contained in the order-L confusability set associated to the channel.
\nThe sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 
\n1) extracting approximately equicoupled subcodes (generalization of equidistant codes) from any using hypergraph Ramsey’s theorem, and 
\n2) significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. 
\n In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which may be of independent interest.
\nOther results include 
\n1) List decoding capacity with asymptotically large L for general adversarial channels; 
\n2) A tight list size bound for most constant composition codes (generalization of constant weight codes); 
\n3) Rederivation and demystification of Blinovsky’s [Blinovsky, 1986] characterization of the list decoding Plotkin points (threshold at which large codes are impossible) for bit-flip channels; 
\n4) Evaluation of general bounds [Wang et al., 2019] for unique decoding in the error correction code setting.

Cited by 2Open access

Secure list decoding

Masahito Hayashi · arXiv (Cornell University) · 2019

In this paper, we propose a new concept of secure list decoding. While the conventional list decoding requires that the list contains the transmitted message, secure list decoding requires the following additional security conditions. The first additional security condition is the impossibility of the correct decoding, i.e., the receiver cannot uniquely identify the transmitted message even though the transmitted message is contained in the list. This condition can be trivially satisfied when the transmission rate is larger than the channel capacity. The other additional security condition is the impossibility for the sender to estimate another element of the decoded list except for the transmitted message. This protocol can be used for anonymous auction, which realizes the anonymity for bidding.

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Generalized List Decoding

Yihan Zhang, Amitalok J. Budkuley, Sidharth Jaggi · arXiv (Cornell University) · 2019

This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip ($\textsf{XOR}$) channels, erasure channels, $\textsf{AND}$ ($Z$-) channels, $\textsf{OR}$ channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) $(L-1)$-list decodable codes (where the list size $L$ is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate $(L-1)$-list decodable codes if and only if the set of completely positive tensors of order-$L$ with admissible marginals is not entirely contained in the order-$L$ confusability set associated to the channel." The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1. extracting equicoupled subcodes (generalization of equidistant code) from any large code sequence using hypergraph Ramsey's theorem, and 2. significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which be may of independent interest. Other results include 1. List decoding capacity with asymptotically large $L$ for general adversarial channels; 2. A tight list size bound for most constant composition codes (generalization of constant weight codes); 3. Rederivation and demystification of Blinovsky's [Bli86] characterization of the list decoding Plotkin points (threshold at which large codes are impossible); 4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error correction code setting.

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PAC Codes: Sequential Decoding vs List Decoding

Mohammad Rowshan, Andreas Burg, Emanuele Viterbo · 2021

In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Arıkan proposed to employ a one-to-one convolutional transform as a pre-coding step before the polar transform. The resulting codes of this concatenation are called polarization-adjusted convolutional (PAC) codes. In this scheme, a pair of polar mapper and demapper as pre- and postprocessing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of the outer code. In this paper, the list decoding and sequential decoding (including Fano decoding and stack decoding) are first adapted for use to decode PAC codes. Then, to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to the reduction of the average decoding time complexity from 50% to 80%, trading with 0.05 to 0.3 dB degradation in error correction performance within FER=10^-3 range, respectively, relative to not applying the corresponding search strategies. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is effective in backtracking and avoids storing the intermediate information or restarting the decoding process. Eventually, all three decoding algorithms are compared in terms of performance, complexity, and resource requirements.

Cited by 7Open access

List decoding using syndromes

Høholdt, Tom, Beelen, Peter · Technical University of Denmark, DTU Orbit (Technical University of Denmark, DTU) · 2008

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List-Decoding Multiplicity Codes.

Swastik Kopparty · 2012

We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching 1 while simultaneously allowing for sublinear-time error correction. In this paper, we show that multiplicity codes also admit powerful and local algorithms that work even in the presence of a large error fraction. In other words, we give algorithms for recovering a polynomial given several evaluations of it and its derivatives, where possibly many of the given evaluations are incorrect. Our first main result shows that univariate multiplicity codes over fields of prime order can be list-decoded up to the so-called list-decoding capacity. Specifically, we show that univariate multiplicity codes of rate R over fields of prime order can be list-decoded from a (1 R e) fraction of errors in polynomial time (for constant R;e). This resembles the behavior of the Folded Reed-Solomon Codes of Guruswami and Rudra (Trans. Info. Theory 2008). The algorithm is based on constructing a differential equation of which the desired codeword is a solution; this differential equation is then solved using a power-series approach (a variation of Hensel lifting) along with other algebraic ideas. Our second main result is a algorithm for decoding multivariate multiplicity codes up to their Johnson radius. The key ingredient of this algorithm is the construction of a special family of algebraically-repelling curves passing through the points of F m ; no moderate-degree multivariate polynomial over F m can simultaneously vanish on all these A version of this paper was posted online as an Electronic Colloq. on Computational Complexity Tech. Report (20). Supported in part by a Sloan Fellowship and NSF grant CCF-1253886.

Cited by 21

On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes

Peter Beelen, Tom Hoholdt, Johan S. R. Nielsen et al. · IEEE Transactions on Information Theory · 2013

We derive the Wu list-decoding algorithm for generalized Reed-Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp-Massey algorithm. We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm, immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes.

Cited by 13Open access

Improved Singleton-type bounds for list-decoding and average-radius list-decoding

Haojie Gu, Hong Lu, Jun Zhang · 2023

List-decoding and average-radius list-decoding are important generalizations of unique decoding that received considerable attention over the years. However, the optimal trade-off among list-decoding radius, list size, and the code rate are not fully understood in both problems. In this paper, firstly, we prove a new Singleton-type bound for list-decodability, which improves the results of [3]. Next, we prove a Singleton-type bound for average-radius list-decodable codes, which is to the best of our knowledge, the first such bound for average-radius list-decodable codes.

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ISSAC software The decoding Library for List Decoding

Guillaume Quintin · 2012

Reed-Solomon (RS) codes form an important and well-studied family of codes. They were first proposed in 1960 by Reed and Solomon in their original paper [RS60]. They are widely used in practice [WB99]. RS codes can be efficiently unique decoded [Gao02] and [Jus76]. Sudan’s 1997 breakthrough on list decoding of RS codes [Sud97], further improved by Guruswami and Sudan in [GS98], showed that RS codes are

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Sequential decoding with biorthogonal alphabet and list decoding

I. M. Jacobs · NASA Technical Reports Server (NASA) · 1965

Cited by 2

List decoding of wavelet codes

D.V. Litichevskii · Izvestiya Instituta Matematiki i Informatiki Udmurt Gos Univ · 2019

Известия Института математики и информатики Удмуртского государственного университета

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List decoding for noisy channels

Peter Eliaš · 1957

Cited by 338

List decoding for noisy channels

DSpace@MIT (Massachusetts Institute of Technology) · 1957

"September 20, 1957." "Reprinted from the 1957 IRE Wescon convention record, part 2."

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Secure codes with list decoding

Yujie Gu, Ilya Vorobyev, Ying Miao · 2022 IEEE International Symposium on Information Theory (ISIT) · 2022

Traitor tracing is a mathematical approach of protecting copyright of multimedia content. In this paper we propose a new concept of secure codes with list decoding (SCLD) for collusion-resistant multimedia fingerprinting, which could include many existing classes of fingerprinting codes as special cases. Furthermore, we build an efficient identifying algorithm for SCLD and establish bounds on its largest asymptotic code rate. In comparison with the existing fingerprinting codes, it is shown that SCLD has not only much more efficient traceability than separable codes but also a much larger code rate than frameproof codes.

Cited by 3

Local list decoding of homomorphisms

Elena Grigorescu · DSpace@MIT (Massachusetts Institute of Technology) · 2006

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.

Cited by 0Open access