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IEEE Communications Letters

Generalized minimum distance iterative decoding of Tanner codes

Saejoon Kim

Published Aug 1, 2005• 1 citation
Publisher

Abstract

In this letter, we present a linear-time decoding algorithm for Tanner codes that can correct errors up to close to half the minimum distance of the code. The algorithm is a simple generalization to the iterative decoding algorithm of expander codes proposed by Skachek and Roth (2003).
DOI
10.1109/lcomm.2005.1496600

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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.
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 thatδd0>2\delta d_{0}>2δd0​>2, corrects up toαn\alpha nαnerrors for a Tanner codeT(G,C0)T\left(G, C_{0}\right)T(G,C0​), whereGGGis a (c,d,α,δc, d, \alpha, \deltac,d,α,δ) bipartite expander withnnnleft vertices, andC0⊆F2dC_{0} \subseteq \mathbb{F}_{2}^{d}C0​⊆F2d​is a linear inner code with minimum distanced0d_{0}d0​. This result improves upon the previous work of Cheng, Ouyang, Shangguan, and Shen (RANDOM 2024), which requiredδd0>3\delta d_{0}>3δd0​>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 ofαn\alpha nαn, compared with the previous radius of2αd0(1+0.5cδ)n\frac{2 \alpha}{d_{0}(1+0.5 c \delta)} nd0​(1+0.5cδ)2α​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 codeT(G,C0)T\left(G, C_{0}\right)T(G,C0​)is approximatelyfδ−1(1d0)αnf_{\delta}^{-1}\left(\frac{1}{d_{0}}\right) \alpha nfδ−1​(d0​1​)αn, wherefδ(⋅)f_{\delta}(\cdot)fδ​(⋅)is the Size-Expansion Function. As another application, we improve the decoding radius of our decoding algorithms fromαn\alpha nαnto approximatelyfδ−1(2d0)αnf_{\delta}^{-1}\left(\frac{2}{d_{0}}\right) \alpha nfδ−1​(d0​2​)αn.
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.
We introduce sequential and parallel decoders for quantum Tanner codes. When the Tanner code construction is applied to a sufficiently expanding square complex with robust local codes, we obtain a family of asymptotically good quantum low-density parity-check codes. In this case, our decoders provably correct arbitrary errors of weight linear in the code length, respectively in linear or logarithmic time. The same decoders are easily adapted to the expander lifted product codes of Panteleev and Kalachev. Along the way, we exploit recently established bounds on the robustness of random tensor codes to give a tighter bound on the minimum distance of quantum Tanner codes.
In this work, we consider iterative soft-decision decoding of binary linear codes using a generalized Tanner graph. A generalized Tanner graph is constructed from the code's Tanner graph representation under the objective of mitigating the effect of small cycles. Then, iterative decoding is applied to the generalized Tanner graph using trellis-based soft-input soft-output decoding in the generalized check nodes. When combined with the stochastic shifting algorithm of Jiang and Narayanan (IEEE Commun. Letters, 2004), the proposed decoding method provides both improved error rate performance and reduced average decoding complexity, without increasing the worst-case decoding complexity.
For binary linear block codes, we introduce a multiple generalized minimum distance (GMD) decoding algorithm, where GMD-like decoding is iterated around a few appropriately selected search centers. Compared with the original GMD decoding by Forney (1966), this decoding algorithm provides better error performance by increasing the number of iterations of erasure and error correction moderately. To reduce the number of iterations, we derive new effective sufficient conditions on the optimality of decoded codewords.