Analysis of Verification-based Decoding on the q-ary Symmetric Channel for Large q

TL;DR

This paper analyzes a verification-based list-message-passing decoder for LDPC codes on the q-ary symmetric channel, deriving density evolution equations, computing thresholds, and comparing algorithms to improve decoding performance for large q.

Contribution

It introduces a detailed analysis of verification-based decoding on q-ary channels, including density evolution, threshold computation, and optimization techniques for large q.

Findings

Decoding thresholds approach capacity for large q with unbounded list sizes.

Maximum list size constraints affect decoding thresholds significantly.

Simulation results confirm the accuracy of density evolution analysis.

Abstract

We discuss and analyze a list-message-passing decoder with verification for low-density parity-check (LDPC) codes on the q-ary symmetric channel (q-SC). Rather than passing messages consisting of symbol probabilities, this decoder passes lists of possible symbols and marks some lists as verified. The density evolution (DE) equations for this decoder are derived and used to compute decoding thresholds. If the maximum list size is unbounded, then we find that any capacity-achieving LDPC code for the binary erasure channel can be used to achieve capacity on the q-SC for large q. The decoding thresholds are also computed via DE for the case where each list is truncated to satisfy a maximum list size constraint. Simulation results are also presented to confirm the DE results. During the simulations, we observed differences between two verification-based decoding algorithms, introduced by Luby and Mitzenmacher, that were implicitly assumed to be identical. In this paper, we provide an analysis of the node-based algorithms from that paper and verify that it matches simulation results. The probability of false verification (FV) is also considered and techniques are discussed to mitigate the FV. Optimization of the degree distribution is also used to improve the threshold for a fixed maximum list size. Finally, the proposed algorithm is compared with a variety of other algorithms using both density evolution thresholds and simulation results.