Improved Decoding of Tanner Codes

TL;DR

This work advances decoding for expander-based Tanner codes by lowering the required expansion product to $\delta d_0 > 2$, and by providing both randomized and deterministic linear-time decoders that correct $\alpha n$ errors. The deterministic decoder is obtained via a derandomization of a prior randomized scheme, preserving the decoding radius and improving the previously known radius bound. A size-expansion framework (via the Size-Expansion Function $f_\delta$) yields new lower bounds for the minimum distance, and enables boosting the decoding radius to $f_\delta^{-1}\left(\frac{2}{d_0}\right)\alpha n$ in refined analyses. Together, these results tighten the connection between expansion properties and decoding performance, and offer practical, scalable decoding for Tanner codes in regimes previously out of reach.

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 $ δd_0 > 2 $, corrects up to $ αn $ errors for a Tanner code $ T(G, C_0) $, where $ G $ is a $ (c, d, α, δ) $-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 $ δd_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 $ αn $, compared with the previous radius of $ \frac{2α}{d_0(1 + 0.5cδ) }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_δ^{-1} \left( \frac{1}{d_0} \right) αn $, where $ f_δ(\cdot) $ is the Size-Expansion Function. As another application, we improve the decoding radius of our decoding algorithms from $αn$ to approximately $f_δ^{-1}\left(\frac{2}{d_0}\right)αn$.

Figures (4)

• Figure 1: The Size-Expansion Function $f_\delta$. For $\delta=0.8$, the trivial bound $\frac{\delta}{k}$ is also shown.
• Figure 2: Plot of $f_\delta^{-1}\left(\frac{1}{d_0}\right)$. For $\delta=0.8$, the factor $\delta d_0$ from the trivial bound $\delta d_0 \alpha n$ is also shown.
• Figure 3: Plot of $f_\delta^{-1}\left(\frac{2}{d_0}\right)$. For $\delta=0.8$, the factor $\frac{\delta d_0}{2}$ from the trivial bound $\frac{\delta d_0}{2} \alpha n$ is also shown.
• Figure 4: An illustration of merging $G_0$ and $G_1$, where $d_0=2$ and $d=3$.

Theorems & Definitions (59)

• Theorem 1.1: Informal version of \ref{['thm:deter_main']}
• Theorem 1.2: Informal version of \ref{['thm:decode_more']}
• Theorem 1.3: Informal version of \ref{['thm:dis_lower', 'thm:dis_upper']}
• Definition 2.1: Bipartite expander
• Definition 2.2: Tanner code
• Definition 2.3: Corrupt bits and unsatisfied checks
• Lemma 2.4: Hoeffding's inequality probcomp
• Lemma 2.5: Azuma's inequality probcomp
• Lemma 2.6
• proof