Non-binary LDPC decoding using truncated messages in the Walsh-Hadamard domain

TL;DR

This paper explores how Walsh-Hadamard transform-based decoders for non-binary LDPC codes can operate on truncated messages to reduce computational complexity, though it does not address the inverse transform.

Contribution

It demonstrates the potential for complexity savings in Walsh-Hadamard based decoders using truncated messages, focusing on the direct transform without inverse transform considerations.

Findings

Significant complexity reduction with truncated messages

Analysis of operation count for non-zero inputs

Highlights need for inverse transform development

Abstract

The Extended Min-Sum (EMS) algorithm for non-binary low-density parity-check (LDPC) defined over an alphabet of size $q$ operates on truncated messages of length $q'$ to achieve a complexity of the order $q'^2$. In contrast, Walsh-Hadamard (WH) transform based iterative decoders achieve a complexity of the order $q\log q$, which is much larger for $q'<<q$. In this paper, we demonstrate that considerable savings can be achieved by letting WH based decoders operate on truncated messages as well. We concentrate on the direct WH transform and compute the number of operations required if only $q'$ of the $q$ inputs are non-zero. Our paper does not cover the inverse WH transform and hence further research is needed to construct WH based decoders that can compete with the EMS algorithm on complexity terms.