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Short Regular Girth-8 QC-LDPC Codes From Exponent Matrices with Vertical Symmetry

Guohua Zhang, Aijing Sun, Ling Liu, Yi Fang

TL;DR

This work introduces vertical symmetry (VS) as a unifying structure for exponent matrices to construct short girth-8 QC-LDPC codes. It provides explicit VS-based constructions for both odd and even column weights and supplements them with a targeted search-based method to further reduce circulant sizes while maintaining decoding performance. For odd weights, it leverages a fundamental VS transformation and sequence-based conditions (including ES and a novel TD sequence) to achieve compact circulants; for even weights, it combines Lemma-driven VS reductions with GCD, DDS, and max-function strategies and a specialized result for J=6. Across extensive simulations, the proposed VS codes deliver shorter circulants with comparable or improved BER/BLER performance relative to state-of-the-art benchmarks, offering practical benefits for low-latency communications in systems such as 6G.

Abstract

To address the challenge of constructing short girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes, a novel construction framework based on vertical symmetry (VS) is proposed. Basic properties of the VS structure are presented. With the aid of these properties, existing explicit constructions for column weights from three to five which can be transformed into the VS structure are sorted out. Then two novel explicit constructions with the VS structure which guarantee short codes are presented for column weights of three and six. Moreover, an efficient search-based method is also proposed to find short codes with the VS structure. Compared with the state-of-the-art benchmarks, both the explicit constructions and the search-based method presented in this paper can provide shorter codes for most cases. Simulation results show that the new shorter codes can perform almost the same as or better than the longer existing counterparts. Thus, the new shorter codes can fit better with the low-latency requirement for modern communication systems.

Short Regular Girth-8 QC-LDPC Codes From Exponent Matrices with Vertical Symmetry

TL;DR

This work introduces vertical symmetry (VS) as a unifying structure for exponent matrices to construct short girth-8 QC-LDPC codes. It provides explicit VS-based constructions for both odd and even column weights and supplements them with a targeted search-based method to further reduce circulant sizes while maintaining decoding performance. For odd weights, it leverages a fundamental VS transformation and sequence-based conditions (including ES and a novel TD sequence) to achieve compact circulants; for even weights, it combines Lemma-driven VS reductions with GCD, DDS, and max-function strategies and a specialized result for J=6. Across extensive simulations, the proposed VS codes deliver shorter circulants with comparable or improved BER/BLER performance relative to state-of-the-art benchmarks, offering practical benefits for low-latency communications in systems such as 6G.

Abstract

To address the challenge of constructing short girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes, a novel construction framework based on vertical symmetry (VS) is proposed. Basic properties of the VS structure are presented. With the aid of these properties, existing explicit constructions for column weights from three to five which can be transformed into the VS structure are sorted out. Then two novel explicit constructions with the VS structure which guarantee short codes are presented for column weights of three and six. Moreover, an efficient search-based method is also proposed to find short codes with the VS structure. Compared with the state-of-the-art benchmarks, both the explicit constructions and the search-based method presented in this paper can provide shorter codes for most cases. Simulation results show that the new shorter codes can perform almost the same as or better than the longer existing counterparts. Thus, the new shorter codes can fit better with the low-latency requirement for modern communication systems.
Paper Structure (18 sections, 4 equations, 5 figures, 4 tables)

This paper contains 18 sections, 4 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Explicit Constructions for Odd Column Weights
  3. VS exponent matrices for $J=3$
  4. VS exponent matrices for $J\geq 5$
  5. Explicit Constructions for Even Column Weights
  6. VS exponent matrices for $J=4$
  7. VS exponent matrices for $J=6$
  8. Search-based method
  9. Governing equations for 4-cycles
  10. Governing equations for 6-cycles
  11. Search algorithm
  12. Performance simulations
  13. Conclusion
  14. Proof of Lemma 1
  15. Proof of Theorem 1
  16. ...and 3 more sections

Figures (5)

  • Figure 1: Circulant size comparison of two explicit VS methods for $J=3$: the proposed TD-based method and existing ES-based method.
  • Figure 2: Circulant size comparison for $J=6$: the proposed explicit VS method and existing random (HS+IRS) method.
  • Figure 3: Performance comparison of $(3,9)$-regular girth-8 VS codes: derived from existing ES and new TD methods.
  • Figure 4: Performance comparison of $(6,11)$-regular girth-8 codes: new VS code ($L=11$ in Theorem 2 (iv)) and existing HS and IRS codes.
  • Figure 5: Performance comparison of $(6,12)$-regular girth-8 codes: new VS code ($L=12$ in Table \ref{['t3']}) and existing IRS code.