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New decoding scheme for LDPC codes based on simple product code structure

Beomkyu Shin, Seokbeom Hong, Hosung Park, Jong-Seon No, Dong-Joon Shin

TL;DR

It is shown that the proposed decoding scheme achieves much better error correcting capability in high SNR region with little additional decoding complexity, compared with the conventional LDPC decoding scheme.

Abstract

In this paper, a new decoding scheme for low-density parity-check (LDPC) codes using the concept of simple product code structure is proposed based on combining two independently received soft-decision data for the same codeword. LDPC codes act as horizontal codes of the product codes and simple algebraic codes are used as vertical codes to help decoding of the LDPC codes. The decoding capability of the proposed decoding scheme is defined and analyzed using the paritycheck matrices of vertical codes and especially the combined-decodability is derived for the case of single parity-check (SPC) and Hamming codes being used as vertical codes. It is also shown that the proposed decoding scheme achieves much better error-correcting capability in high signal to noise ratio (SNR) region with low additional decoding complexity, compared with a conventional decoding scheme.

New decoding scheme for LDPC codes based on simple product code structure

TL;DR

It is shown that the proposed decoding scheme achieves much better error correcting capability in high SNR region with little additional decoding complexity, compared with the conventional LDPC decoding scheme.

Abstract

In this paper, a new decoding scheme for low-density parity-check (LDPC) codes using the concept of simple product code structure is proposed based on combining two independently received soft-decision data for the same codeword. LDPC codes act as horizontal codes of the product codes and simple algebraic codes are used as vertical codes to help decoding of the LDPC codes. The decoding capability of the proposed decoding scheme is defined and analyzed using the paritycheck matrices of vertical codes and especially the combined-decodability is derived for the case of single parity-check (SPC) and Hamming codes being used as vertical codes. It is also shown that the proposed decoding scheme achieves much better error-correcting capability in high signal to noise ratio (SNR) region with low additional decoding complexity, compared with a conventional decoding scheme.

Paper Structure

This paper contains 12 sections, 9 theorems, 25 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A New Decoding Scheme Based on Simple Product Code Structure
  3. Notations and Basic Operations
  4. A New Decoding Scheme
  5. Examples
  6. SPC Codes as Vertical Codes
  7. $(7,4)$ Hamming Codes as Vertical Codes
  8. Combined-Decodability of the Proposed Decoding Scheme
  9. Definitions and Properties of Combined-Decodability
  10. Combined-Decodability of Hamming Codes
  11. Simulation Results
  12. Conclusion

Key Result

Lemma 3

If a vertical code $\mathcal{C}$ with a parity-check matrix $H$ is $\epsilon$ combinable, any vertical code $\tilde{\mathcal{C}}$ whose parity-check matrix $\tilde{H}$ is constructed by selecting $\epsilon$ or more columns from $H$ is also $\epsilon$ combinable.

Figures (3)

  • Figure 1: Structure of a codeword matrix of product code.
  • Figure 2: Flowchart of the proposed decoding process.
  • Figure 3: Performance comparison of the conventional decoding of LDPC codes in IEEE802.16e and the proposed decoding with $(24,23)$ SPC code as a vertical code.

Theorems & Definitions (18)

  • Definition 1: $\epsilon$ combinable
  • Definition 2: $\eta$ combined-decodable
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • Lemma 7
  • ...and 8 more