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arXiv:cs/0508062

Decoding of Expander Codes at Rates Close to Capacity

Alexei Ashikhmin, Vitaly Skachek

Published Aug 12, 2005• 1 citation
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Abstract

The decoding error probability of codes is studied as a function of their block length. It is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability. Specifically, it is assumed that there exists a family of codes of length N and rate R=(1-ε)C (C is a capacity of a binary symmetric channel), whose decoding probability decreases polynomially in 1/N. It is shown that if the decoding probability decreases sufficiently fast, but still only polynomially fast in 1/N, then there exists another such family of codes whose decoding error probability decreases exponentially fast in N. Moreover, if the decoding time complexity of the assumed family of codes is polynomial in N and 1/ε, then the decoding time complexity of the presented family is linear in N and polynomial in 1/ε. These codes are compared to the recently presented codes of Barg and Zemor, ``Error Exponents of Expander Codes,'' IEEE Trans. Inform. Theory, 2002, and ``Concatenated Codes: Serial and Parallel,'' IEEE Trans. Inform. Theory, 2005. It is shown that the latter families can not be tuned to have exponentially decaying (in N) error probability, and at the same time to have decoding time complexity linear in N and polynomial in 1/ε.

DOI
10.48550/arxiv.cs/0508062

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The decoding error probability of codes is studied as a function of their block length. It is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability. Specifically, it is assumed that there exists a family of codes of length N and rate R=(1-ε)C (C is a capacity of a binary symmetric channel), whose decoding probability decreases polynomially in 1/N. It is shown that if the decoding probability decreases sufficiently fast, but still only polynomially fast in 1/N, then there exists another such family of codes whose decoding error probability decreases exponentially fast in N. Moreover, if the decoding time complexity of the assumed family of codes is polynomial in N and 1/ε, then the decoding time complexity of the presented family is linear in N and polynomial in 1/ε. These codes are compared to the recently presented codes of Barg and Zemor, Error Exponents of Expander Codes,'' IEEE Trans. Inform. Theory, 2002, and Concatenated Codes: Serial and Parallel,'' IEEE Trans. Inform. Theory, 2005. It is shown that the latter families can not be tuned to have exponentially decaying (in N) error probability, and at the same time to have decoding time complexity linear in N and polynomial in 1/ε.
The decoding error probability of codes is studied as a function of their block length. It is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability. Specifically, it is assumed that there exists a family of codes of length N and rate R=(1-epsiv)C (C is a capacity of a binary-symmetric channel), whose decoding probability decreases inverse polynomially in N. It is shown that if the decoding probability decreases sufficiently fast, but still only inverse polynomially fast in N, then there exists another such family of codes whose decoding error probability decreases exponentially fast in N. Moreover, if the decoding time complexity of the assumed family of codes is polynomial in N and 1/epsiv, then the decoding time complexity of the presented family is linear in N and polynomial in 1/epsiv. These codes are compared to the recently presented codes of Barg and Zemor, "Error Exponents of Expander Codes", IEEE Transactions on Information Theory, 2002, and "Concatenated Codes: Serial and Parallel", IEEE Transactions on Information Theory, 2005. It is shown that the latter families cannot be tuned to have exponentially decaying (in N) error probability, and at the same time to have decoding time complexity linear in N and polynomial in 1/epsiv
The concatenation of nearly-MDS expander codes of Roth and Skachek, "On Nearly-MDS Expander Codes," Proc. IEEE ISIT'04, with 'typical' LDPC codes is investigated. It is shown that for the rates R = (1 - epsi)C (C is the capacity of the binary symmetric channel (BSC)), under certain condition on the parameters of LDPC codes, these concatenated codes have decoding time linear in their length and polynomial in 1/epsi, and the decoding error probability decays exponentially. These codes are compared to the recently presented codes of Barg and Zemor, "Error Exponents of Expander Codes," IEEE Trans. Inform. Theory, 2002, and "Concatenated Codes: Serial and Parallel," IEEE Trans. Inform. Theory, 2005. It is shown that the latter families can not be tuned to have all the aforementioned properties
This paper deals with the specific construction of binary low-density parity-check (LDPC) codes. We derive lower bounds on the error exponents for these codes transmitted over the memoryless binary symmetric channel (BSC) for both the well-known maximum-likelihood (ML) and proposed low-complexity decoding algorithms. We prove the existence of such LDPC codes that the probability of erroneous decoding decreases exponentially with the growth of the code length while keeping coding rates below the corresponding channel capacity. We also show that an obtained error exponent lower bound under ML decoding almost coincide with the error exponents of good linear codes.
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.