Improved Decoding of Tanner Codes

Abstract: 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(G, C_0) $, 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 + 0.5c\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 $T(G,C_0)$ 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 $f_\delta^{-1}\left(\frac{2}{d_0}\right)\alpha n$.