arXiv:2002.08579 · IEEE International Symposium on Information Theory (ISIT)

Linear-Time Erasure List-Decoding of Expander Codes

Noga Ron‐Zewi, Mary K. Wootters, Gilles Zémor

Published Jun 1, 2020• 6 citations

Abstract

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r > 0 be any integer. Given an inner codeC0of length d, and a d-regular bipartite expander graph G with n vertices on each side, we give an algorithm to list-decode the codeC=C(G,C0) of length nd from approximately δδrnd erasures in time n·poly (d2r/δ), where δ and δrare the relative distance and the r'th generalized relative distance ofC0, respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately δ2nd. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on r and δ; then we show how to improve the dependence of the running time on these parameters.

DOI: 10.1109/isit44484.2020.9174325

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Linear-time decoding from a constant fraction of errors for irregular expander codes

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When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

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When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

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arXiv:2002.08579 · IEEE International Symposium on Information Theory (ISIT)

Linear-Time Erasure List-Decoding of Expander Codes

Noga Ron‐Zewi, Mary K. Wootters, Gilles Zémor

Published Jun 1, 2020• 6 citations

arXiv HTML PDF

Abstract

DOI: 10.1109/isit44484.2020.9174325

Related papers

Fast Decoding of Expander Codes

Michael C. Dowling, Shuhong Gao

Expander codes are Tanner codes defined on sparse graphs that have good expansion properties. Sipser and Spielman (1996) showed that there is a linear-time decoding algorithm for expander codes when the vertex expansion is at least 3/4 and the number of errors corrected is a constant fraction of the code length. Later, Feldman et al. (2007) gave a decoding algorithm that allows the expansion to be 2/3 + 1/(3c), where $c$ is the left degree of the underlying bipartite graph, at the expense of polynomial-time decoding complexity. Recently, Viderman (2013) further improved the expansion parameter to $2/3 - 1/(6c)$ , and the decoding algorithm runs in linear time. These results are for expander codes whose inner codes are parity-check codes. By using stronger inner codes, Chilappagari et al. (2010) showed that there is a linear-time decoding algorithm for every vertex expansion greater than 1/2. In this paper, it is shown that for every vertex expansion, there is a linear-time decoding algorithm for expander codes (using inner codes with minimum distance depending on the vertex expansion), and that the number of errors corrected is a constant fraction of the code length.Expander codes are Tanner codes defined on sparse graphs that have good expansion properties. Sipser and Spielman (1996) showed that there is a linear-time decoding algorithm for expander codes when the vertex expansion is at least 3/4 and the number of errors corrected is a constant fraction of the code length. Later, Feldman et al. (2007) gave a decoding algorithm that allows the expansion to be 2/3 + 1/(3c), where $c$ is the left degree of the underlying bipartite graph, at the expense of polynomial-time decoding complexity. Recently, Viderman (2013) further improved the expansion parameter to $2/3 - 1/(6c)$ , and the decoding algorithm runs in linear time. These results are for expander codes whose inner codes are parity-check codes. By using stronger inner codes, Chilappagari et al. (2010) showed that there is a linear-time decoding algorithm for every vertex expansion greater than 1/2. In this paper, it is shown that for every vertex expansion, there is a linear-time decoding algorithm for expander codes (using inner codes with minimum distance depending on the vertex expansion), and that the number of errors corrected is a constant fraction of the code length.

IEEE Transactions on Information TheoryJul 12, 20174 citations

Improved Decoding of Tanner Codes

Zhaienhe Zhou, Zeyu Guo

In this paper, we present improved decoding algorithms for expander-based Tanner codes. We begin by developing a randomized linear-time decoding algorithm that, under the condition that $\delta d_{0}>2$ , corrects up to $\alpha n$ errors for a Tanner code $T\left(G, C_{0}\right)$ , where $G$ is a ( $c, d, \alpha, \delta$ ) bipartite expander with $n$ left vertices, and $C_{0} \subseteq \mathbb{F}_{2}^{d}$ is a linear inner code with minimum distance $d_{0}$ . This result improves upon the previous work of Cheng, Ouyang, Shangguan, and Shen (RANDOM 2024), which required $\delta d_{0}>3$ . We further derandomize the algorithm to obtain a deterministic linear-time decoding algorithm with the same decoding radius. Our algorithm improves upon the previous deterministic algorithm of Cheng et al. by achieving a decoding radius of $\alpha n$ , compared with the previous radius of $\frac{2 \alpha}{d_{0}(1+0.5 c \delta)} n$ . Additionally, we investigate the size-expansion trade-off introduced by the recent work of Chen, Cheng, Li, and Ouyang (IEEE TIT 2023), and use it to provide new bounds on the minimum distance of Tanner codes. Specifically, we prove that the minimum distance of a Tanner code $T\left(G, C_{0}\right)$ is approximately $f_{\delta}^{-1}\left(\frac{1}{d_{0}}\right) \alpha n$ , where $f_{\delta}(\cdot)$ is the Size-Expansion Function. As another application, we improve the decoding radius of our decoding algorithms from $\alpha n$ to approximately $f_{\delta}^{-1}\left(\frac{2}{d_{0}}\right) \alpha n$ .In this paper, we present improved decoding algorithms for expander-based Tanner codes. We begin by developing a randomized linear-time decoding algorithm that, under the condition that$\delta d_{0}>2$, corrects up to$\alpha n$errors for a Tanner code$T\left(G, C_{0}\right)$, where$G$is a ($c, d, \alpha, \delta$) bipartite expander with$n$left vertices, and$C_{0} \subseteq \mathbb{F}_{2}^{d}$is a linear inner code with minimum distance$d_{0}$. This result improves upon the previous work of Cheng, Ouyang, Shangguan, and Shen (RANDOM 2024), which required$\delta d_{0}>3$. We further derandomize the algorithm to obtain a deterministic linear-time decoding algorithm with the same decoding radius. Our algorithm improves upon the previous deterministic algorithm of Cheng et al. by achieving a decoding radius of$\alpha n$, compared with the previous radius of$\frac{2 \alpha}{d_{0}(1+0.5 c \delta)} n$. Additionally, we investigate the size-expansion trade-off introduced by the recent work of Chen, Cheng, Li, and Ouyang (IEEE TIT 2023), and use it to provide new bounds on the minimum distance of Tanner codes. Specifically, we prove that the minimum distance of a Tanner code$T\left(G, C_{0}\right)$is approximately$f_{\delta}^{-1}\left(\frac{1}{d_{0}}\right) \alpha n$, where$f_{\delta}(\cdot)$is the Size-Expansion Function. As another application, we improve the decoding radius of our decoding algorithms from$\alpha n$to approximately$f_{\delta}^{-1}\left(\frac{2}{d_{0}}\right) \alpha n$.

arXiv:2501.12293IEEE International Symposium on Information Theory (ISIT)Jun 22, 20252 citations

Linear-time decoding from a constant fraction of errors for irregular expander codes

Rutuja Kshirsagar, Gretchen L. Matthews

In this paper, we present a linear-time decoding algorithm for expander codes based on irregular graphs of any positive vertex expansion factor [Formula: see text] and inner codes with a minimum distance of at least [Formula: see text], where [Formula: see text] is the maximum right degree. The algorithm corrects a constant fraction of errors. It builds on two thrusts. The first is a series of works starting with that of Sipser and Spielman [Expander codes, IEEE Trans. Inf. Theory 42(6) (1996) 1710–1722] demonstrating that an asymptotically good family of error-correcting codes that can be decoded in linear time even from a constant fraction of errors in a received word provided [Formula: see text] is at least [Formula: see text] and continuing to the results of Gao and Dowling [Fast decoding of expander codes, IEEE Trans. Inf. Theory 64(2) (2018) 972–978], which only require [Formula: see text] provided the inner code minimum distance is sufficiently large. The second is the improved performance of LDPC codes based on irregular graphs demonstrated by Luby et al. [Improved low- density parity-check codes using irregular graphs, IEEE Trans. Inf. Theory 47(2) (2001) 585–598] and Richardson et al. [Design of capacity- approaching irregular low-density parity-check codes, IEEE Trans. Inf. Theory 47(2) (2001) 619–637]. The algorithm presented in this paper allows for irregular or regular graph-based constructions and uses inner codes of appropriate lengths as checks rather than simple parity-checks.In this paper, we present a linear-time decoding algorithm for expander codes based on irregular graphs of any positive vertex expansion factor [Formula: see text] and inner codes with a minimum distance of at least [Formula: see text], where [Formula: see text] is the maximum right degree. The algorithm corrects a constant fraction of errors. It builds on two thrusts. The first is a series of works starting with that of Sipser and Spielman [Expander codes, IEEE Trans. Inf. Theory 42(6) (1996) 1710–1722] demonstrating that an asymptotically good family of error-correcting codes that can be decoded in linear time even from a constant fraction of errors in a received word provided [Formula: see text] is at least [Formula: see text] and continuing to the results of Gao and Dowling [Fast decoding of expander codes, IEEE Trans. Inf. Theory 64(2) (2018) 972–978], which only require [Formula: see text] provided the inner code minimum distance is sufficiently large. The second is the improved performance of LDPC codes based on irregular graphs demonstrated by Luby et al. [Improved low- density parity-check codes using irregular graphs, IEEE Trans. Inf. Theory 47(2) (2001) 585–598] and Richardson et al. [Design of capacity- approaching irregular low-density parity-check codes, IEEE Trans. Inf. Theory 47(2) (2001) 619–637]. The algorithm presented in this paper allows for irregular or regular graph-based constructions and uses inner codes of appropriate lengths as checks rather than simple parity-checks.

Journal of Algebra and Its ApplicationsApr 16, 20250 citations

Explicit Capacity-Achieving Quantum LDPC Codes List Decodable in Near-linear Time

William Gay, Fernando Granha Jeronimo, Abhi Shukul

In classical coding theory, the quest for explicit codes achieving list decoding capacity has been an important driving force. While random codes are easily shown to achieve capacity, an explicit construction was only discovered decades later in the seminal work [Guruswami and Rudra, STOC 2006]. In quantum coding theory, it is highly desirable that a code family be LDPC. While good quantum codes were known for decades, obtaining the additional LDPC property was elusive. In fact, only very recently that good quantum LDPC codes were discovered in a breakthrough work [Panteleev and Kalachev, STOC 2022]. In this context, a natural question is to ask for an explicit family of quantum codes achieving list decoding capacity while also possessing the important LDPC property. In this work, we provide (to the best of our knowledge) the first explicit constructions of quantum LDPC codes achieving list decoding capacity, namely, with a list decoding radius approaching the quantum Singleton bound with constant list sizes. Furthermore, we provide near-linear time (in the block-length) list decoding algorithms approaching capacity. Our explicit code construction are based on expander graphs via the quantum analogue of Alon-Edmonds-Luby (AEL) amplification [Bergamaschi, Golowich and Gunn, STOC 2024], and this enables the important LDPC property. Our efficient list decoding algorithms are obtained by generalizing the classical expander-based weak-regularity list decoding algorithms [Srivastava and Tulsiani, FOCS 2025] [Jeronimo and Singh, 2025] to suitable instantiations of quantum AEL.In classical coding theory, the quest for explicit codes achieving list decoding capacity has been an important driving force. While random codes are easily shown to achieve capacity, an explicit construction was only discovered decades later in the seminal work [Guruswami and Rudra, STOC 2006]. In quantum coding theory, it is highly desirable that a code family be LDPC. While good quantum codes were known for decades, obtaining the additional LDPC property was elusive. In fact, only very recently that good quantum LDPC codes were discovered in a breakthrough work [Panteleev and Kalachev, STOC 2022]. In this context, a natural question is to ask for an explicit family of quantum codes achieving list decoding capacity while also possessing the important LDPC property. In this work, we provide (to the best of our knowledge) the first explicit constructions of quantum LDPC codes achieving list decoding capacity, namely, with a list decoding radius approaching the quantum Singleton bound with constant list sizes. Furthermore, we provide near-linear time (in the block-length) list decoding algorithms approaching capacity. Our explicit code construction are based on expander graphs via the quantum analogue of Alon-Edmonds-Luby (AEL) amplification [Bergamaschi, Golowich and Gunn, STOC 2024], and this enables the important LDPC property. Our efficient list decoding algorithms are obtained by generalizing the classical expander-based weak-regularity list decoding algorithms [Srivastava and Tulsiani, FOCS 2025] [Jeronimo and Singh, 2025] to suitable instantiations of quantum AEL.

quant-pharXiv:2609.40313Oct 1, 20260 citations

When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

Yuanting Shen, Chong Shangguan, Minghui Ouyang +1 more

Tanner codes are error-correcting codes built from a bipartite graphGand a short inner codeC0. Expander codes are a special type of Tanner code, where the graph is highly interconnected, ensuring stronger error correction capabilities. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that δ ∈ [0, 1] andd0∈ N must satisfy, so thateverybipartite expanderGwith vertex expansion ratio δ andeverylinear inner codeC0with minimum distanced0together define an expander code that corrects Ω(n) errors inO(n) time? ForC0being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT’96) showed that δ > 3/4 is sufficient; later, Viderman (ACM-TOCT’13) improved this to δ > 2/3 - Ω(1) and he also showed that δ > 1/2 is necessary. For general linear codeC0, the previously best-known result of Dowling and Gao (IEEE-TIT’18) showed thatd0= Ω(cδ-2) is sufficient, wherecis the left-degree ofG. We present a near-optimal solution to the above problem for generalC0by showing that δd0> 3 is sufficient and δd0> 1 is necessary, thereby significantly improving Dowling-Gao’s result. To prove the sufficient condition, we present two novel algorithms for decoding arbitrary expander codes withδd0> 3, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius. To prove the necessary condition, we generalize the aforementioned necessary result of Viderman, and construct for every pair of δ,d0with δd0=1, an expander code with constant distance, that only corrects a constant number of errors.Tanner codes are error-correcting codes built from a bipartite graphGand a short inner codeC0. Expander codes are a special type of Tanner code, where the graph is highly interconnected, ensuring stronger error correction capabilities. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that δ ∈ [0, 1] andd0∈ N must satisfy, so thateverybipartite expanderGwith vertex expansion ratio δ andeverylinear inner codeC0with minimum distanced0together define an expander code that corrects Ω(n) errors inO(n) time? ForC0being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT’96) showed that δ > 3/4 is sufficient; later, Viderman (ACM-TOCT’13) improved this to δ > 2/3 - Ω(1) and he also showed that δ > 1/2 is necessary. For general linear codeC0, the previously best-known result of Dowling and Gao (IEEE-TIT’18) showed thatd0= Ω(cδ-2) is sufficient, wherecis the left-degree ofG. We present a near-optimal solution to the above problem for generalC0by showing that δd0> 3 is sufficient and δd0> 1 is necessary, thereby significantly improving Dowling-Gao’s result. To prove the sufficient condition, we present two novel algorithms for decoding arbitrary expander codes withδd0> 3, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius. To prove the necessary condition, we generalize the aforementioned necessary result of Viderman, and construct for every pair of δ,d0with δd0=1, an expander code with constant distance, that only corrects a constant number of errors.

IEEE Transactions on Information TheoryJul 7, 20250 citations

When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

Kuan Cheng, Minghui Ouyang, Chong Shangguan +1 more

Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph $G$ together with a linear inner code $C_0$ . Expander codes are Tanner codes whose defining bipartite graph $G$ has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that $δ$ and $d_0$ must satisfy, so that \textit{every} bipartite expander $G$ with vertex expansion ratio $δ$ and \textit{every} linear inner code $C_0$ with minimum distance $d_0$ together define an expander code that corrects $Ω(n)$ errors in $O(n)$ time? For $C_0$ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that $δ>3/4$ is sufficient; later Viderman (ACM-TOCT'13) improved this to $δ>2/3-Ω(1)$ and he also showed that $δ>1/2$ is necessary. For general linear code $C_0$ , the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that $d_0=Ω(cδ^{-2})$ is sufficient, where $c$ is the left-degree of $G$ . In this paper, we give a near-optimal solution to the above question for general $C_0$ by showing that $δd_0>3$ is sufficient and $δd_0>1$ is necessary, thereby also significantly improving Dowling-Gao's result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph $G$ together with a linear inner code $C_0$. Expander codes are Tanner codes whose defining bipartite graph $G$ has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that $δ$ and $d_0$ must satisfy, so that \textit{every} bipartite expander $G$ with vertex expansion ratio $δ$ and \textit{every} linear inner code $C_0$ with minimum distance $d_0$ together define an expander code that corrects $Ω(n)$ errors in $O(n)$ time? For $C_0$ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that $δ>3/4$ is sufficient; later Viderman (ACM-TOCT'13) improved this to $δ>2/3-Ω(1)$ and he also showed that $δ>1/2$ is necessary. For general linear code $C_0$, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that $d_0=Ω(cδ^{-2})$ is sufficient, where $c$ is the left-degree of $G$. In this paper, we give a near-optimal solution to the above question for general $C_0$ by showing that $δd_0>3$ is sufficient and $δd_0>1$ is necessary, thereby also significantly improving Dowling-Gao's result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.

arXiv:2312.16087Leibniz international proceedings in informaticsDec 26, 20230 citations

c

is the left degree of the underlying bipartite graph, at the expense of polynomial-time decoding complexity. Recently, Viderman (2013) further improved the expansion parameter to

2/3 - 1/(6c)

, and the decoding algorithm runs in linear time. These results are for expander codes whose inner codes are parity-check codes. By using stronger inner codes, Chilappagari et al. (2010) showed that there is a linear-time decoding algorithm for every vertex expansion greater than 1/2. In this paper, it is shown that for every vertex expansion, there is a linear-time decoding algorithm for expander codes (using inner codes with minimum distance depending on the vertex expansion), and that the number of errors corrected is a constant fraction of the code length.

In this paper, we present improved decoding algorithms for expander-based Tanner codes. We begin by developing a randomized linear-time decoding algorithm that, under the condition that

\delta d_{0}>2

, corrects up to

\alpha n

errors for a Tanner code

T\left(G, C_{0}\right)

, where

G

is a (

c, d, \alpha, \delta

) bipartite expander with

n

left vertices, and

C_{0} \subseteq \mathbb{F}_{2}^{d}

is a linear inner code with minimum distance

d_{0}

. This result improves upon the previous work of Cheng, Ouyang, Shangguan, and Shen (RANDOM 2024), which required

\delta d_{0}>3

. We further derandomize the algorithm to obtain a deterministic linear-time decoding algorithm with the same decoding radius. Our algorithm improves upon the previous deterministic algorithm of Cheng et al. by achieving a decoding radius of

\alpha n

, compared with the previous radius of

\frac{2 \alpha}{d_{0}(1+0.5 c \delta)} n

. Additionally, we investigate the size-expansion trade-off introduced by the recent work of Chen, Cheng, Li, and Ouyang (IEEE TIT 2023), and use it to provide new bounds on the minimum distance of Tanner codes. Specifically, we prove that the minimum distance of a Tanner code

T\left(G, C_{0}\right)

is approximately

f_{\delta}^{-1}\left(\frac{1}{d_{0}}\right) \alpha n

, where

f_{\delta}(\cdot)

is the Size-Expansion Function. As another application, we improve the decoding radius of our decoding algorithms from

\alpha n

to approximately

f_{\delta}^{-1}\left(\frac{2}{d_{0}}\right) \alpha n

Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph

G

together with a linear inner code

C_0

. Expander codes are Tanner codes whose defining bipartite graph

G

has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that

δ

and

d_0

must satisfy, so that \textit{every} bipartite expander

G

with vertex expansion ratio

δ

and \textit{every} linear inner code

C_0

with minimum distance

d_0

together define an expander code that corrects

Ω(n)

errors in

O(n)

time? For

C_0

being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that

δ>3/4

is sufficient; later Viderman (ACM-TOCT'13) improved this to

δ>2/3-Ω(1)

and he also showed that

δ>1/2

is necessary. For general linear code

C_0

, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that

d_0=Ω(cδ^{-2})

is sufficient, where

c

is the left-degree of

G

. In this paper, we give a near-optimal solution to the above question for general

C_0

by showing that

δd_0>3

is sufficient and

δd_0>1

is necessary, thereby also significantly improving Dowling-Gao's result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.