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Decoding of NB-LDPC codes over Subfields

V. B. Wijekoon, Emanuele Viterbo, Yi Hong

TL;DR

This approach offers several decoding strategies for a single NB-LDPC code, with varying levels of performance-complexity trade-offs, based on a novel method of expanding a non-binary Tanner graphs over a finite field into a graph over a subfield.

Abstract

The non-binary low-density parity-check (NB-LDPC) codes can offer promising performance advantages but suffer from high decoding complexity. To tackle this challenge, in this paper, we consider NB-LDPC codes over finite fields as codes over \textit{subfields} as a means of reducing decoding complexity. In particular, our approach is based on a novel method of expanding a non-binary Tanner graph over a finite field into a graph over a subfield. This approach offers several decoding strategies for a single NB-LDPC code, with varying levels of performance-complexity trade-offs. Simulation results demonstrate that in a majority of cases, performance loss is minimal when compared with the complexity gains.

Decoding of NB-LDPC codes over Subfields

TL;DR

This approach offers several decoding strategies for a single NB-LDPC code, with varying levels of performance-complexity trade-offs, based on a novel method of expanding a non-binary Tanner graphs over a finite field into a graph over a subfield.

Abstract

The non-binary low-density parity-check (NB-LDPC) codes can offer promising performance advantages but suffer from high decoding complexity. To tackle this challenge, in this paper, we consider NB-LDPC codes over finite fields as codes over \textit{subfields} as a means of reducing decoding complexity. In particular, our approach is based on a novel method of expanding a non-binary Tanner graph over a finite field into a graph over a subfield. This approach offers several decoding strategies for a single NB-LDPC code, with varying levels of performance-complexity trade-offs. Simulation results demonstrate that in a majority of cases, performance loss is minimal when compared with the complexity gains.

Paper Structure

This paper contains 9 sections, 5 theorems, 17 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. $\alpha$-connected Subgroups
  3. Graph Expansion
  4. Preliminaries
  5. Graph Expansion
  6. Decoding Scheme
  7. Simulation Results
  8. Decoding Complexity
  9. Conclusions

Key Result

Lemma 1

Consider $\mathbb{F}_{p^r}$ and let $m~|~r$. Then the smallest possible $\alpha$-connected set of additive subgroups of order $p^{r-m}$ has a cardinality of $\frac{p^r-1}{p^m-1}$.

Figures (4)

  • Figure 1: Initial Expansion
  • Figure 2: Final Expansion
  • Figure 3: FER Perf. with a (1998,1776) code over $GF(2^6)$ ($\mathcal{C}_1$)
  • Figure 4: FER Perf. with a (1000,861) code over $GF(2^4)$ ($\mathcal{C}_2$)

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 3
  • Lemma 4
  • ...and 4 more