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DIMACS series in discrete mathematics and theoretical computer science

Multilevel expander codes

Alexander Barg, Gilles Zémor

Published Jul 19, 2005• 1 citation
Publisher

Abstract

We define multilevel codes on bipartite graphs which have properties analogous to multilevel serial concatenations. A linear-time decoding algorithm is described that corrects a proportion of errors equal to half the Blokh-Zyablov bound. The error probability of this algorithm has exponent similar to that of serially concatenated multilevel codes, i.e. equals the best-known exponent achievable by a polynomial-time decoding algorithm

DOI
10.1090/dimacs/068/04

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We define multilevel codes on bipartite graphs which have properties analogous to multilevel serial concatenations. A linear-time decoding algorithm is described that corrects a proportion of errors equal to half the Blokh-Zyablov bound. The error probability of this algorithm has exponent similar to that of serially concatenated multilevel codes, i.e. equals the best-known exponent achievable by a polynomial-time decoding algorithm
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
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 ccc 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)2/3 - 1/(6c)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.
Recently, G. Zemor (see IEEE Trans. Inf. Theory, vol.47, p.835-7, 2001) proposed an improvement on the Sipser-Spielman analysis of expander codes (Sipser, M. and Spielman, D.A., IEEE Trans. Inf. Theory, vol.42 , p.1710-22, 1996) and presented a linear-time iterative decoder that can correct a number of errors up to approximately 1/4 the known lower bound on the minimum distance of the code. We propose an improvement on Zemor's decoder for F=GF(2), with the number of correctable errors becoming close to half the lower bound on the minimum distance. The improvement is obtained by inserting into the decoding algorithm features akin to generalized minimum distance decoding of concatenated codes.
Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph GGG together with a linear inner code C0C_0C0​. Expander codes are Tanner codes whose defining bipartite graph GGG 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 δδδ and d0d_0d0​ must satisfy, so that \textit{every} bipartite expander GGG with vertex expansion ratio δδδ and \textit{every} linear inner code C0C_0C0​ with minimum distance d0d_0d0​ together define an expander code that corrects Ω(n)Ω(n)Ω(n) errors in O(n)O(n)O(n) time? For C0C_0C0​ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that δ>3/4δ>3/4δ>3/4 is sufficient; later Viderman (ACM-TOCT'13) improved this to δ>2/3−Ω(1)δ>2/3-Ω(1)δ>2/3−Ω(1) and he also showed that δ>1/2δ>1/2δ>1/2 is necessary. For general linear code C0C_0C0​, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that d0=Ω(cδ−2)d_0=Ω(cδ^{-2})d0​=Ω(cδ−2) is sufficient, where ccc is the left-degree of GGG. In this paper, we give a near-optimal solution to the above question for general C0C_0C0​ by showing that δd0>3δd_0>3δd0​>3 is sufficient and δd0>1δd_0>1δd0​>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.