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On Existence of Girth-8 QC-LDPC Code with Large Column Weight: Combining Mirror-sequence with Classification Modulo Ten

Guohua Zhang, Xiangya Liu, Jianhua Zhang, Yi Fang

TL;DR

This work develops algebraic QC-LDPC code constructions with girth 8 and large column weights by introducing a mirror sequence and a modulo-10 row-regrouping scheme. For J=7 and J=8 (row weights), the authors provide explicit constructions and derived variants, prove absence of 4- and 6-cycles via GCD-triple analysis, and establish lower bounds on consecutive circulant sizes that scale as $P_{LB}=4(L-1)(L+5)+1$ (J=7) and $P_{LB}=4(L-1)(L+7)+1$ (J=8). They also show circulant sizes smaller than these bounds are achievable through P1/P2-type arguments, yielding upper bounds $P_s \,\le\, (L+1)(3L+13)$ (J=7) and $P_s \,\le\, (3L+4)(L+6)+7$ (J=8), with asymptotic improvements of about 20% in lower bounds and ~25% smaller sizes below bounds. Compared to existing benchmarks, these algebraic constructions deliver shorter lengths for the same girth and provide practical performance gains relative to symmetry-based search methods. The results have implications for channel coding, compressed sensing, and distributed storage by enabling compact QC-LDPC codes with robust girth properties.

Abstract

Quasi-cyclic (QC) LDPC codes with large girths play a crucial role in several research and application fields, including channel coding, compressed sensing and distributed storage systems. A major challenge in respect of the code construction is how to obtain such codes with the shortest possible length (or equivalently, the smallest possible circulant size) using algebraic methods instead of search methods. The greatest-common-divisor (GCD) framework we previously proposed has algebraically constructed QC-LDPC codes with column weights of 5 and 6, very short lengths, and a girth of 8. By introducing the concept of a mirror sequence and adopting a new row-regrouping scheme, QC-LDPC codes with column weights of 7 and 8, very short lengths, and a girth of 8 are proposed for arbitrary row weights in this article via an algebraic manner under the GCD framework. Thanks to these novel algebraic methods, the lower bounds (for column weights 7 and 8) on consecutive circulant sizes are both improved by asymptotically about 20%, compared with the existing benchmarks. Furthermore, these new constructions can also offer circulant sizes asymptotically about 25% smaller than the novel bounds.

On Existence of Girth-8 QC-LDPC Code with Large Column Weight: Combining Mirror-sequence with Classification Modulo Ten

TL;DR

This work develops algebraic QC-LDPC code constructions with girth 8 and large column weights by introducing a mirror sequence and a modulo-10 row-regrouping scheme. For J=7 and J=8 (row weights), the authors provide explicit constructions and derived variants, prove absence of 4- and 6-cycles via GCD-triple analysis, and establish lower bounds on consecutive circulant sizes that scale as (J=7) and (J=8). They also show circulant sizes smaller than these bounds are achievable through P1/P2-type arguments, yielding upper bounds (J=7) and (J=8), with asymptotic improvements of about 20% in lower bounds and ~25% smaller sizes below bounds. Compared to existing benchmarks, these algebraic constructions deliver shorter lengths for the same girth and provide practical performance gains relative to symmetry-based search methods. The results have implications for channel coding, compressed sensing, and distributed storage by enabling compact QC-LDPC codes with robust girth properties.

Abstract

Quasi-cyclic (QC) LDPC codes with large girths play a crucial role in several research and application fields, including channel coding, compressed sensing and distributed storage systems. A major challenge in respect of the code construction is how to obtain such codes with the shortest possible length (or equivalently, the smallest possible circulant size) using algebraic methods instead of search methods. The greatest-common-divisor (GCD) framework we previously proposed has algebraically constructed QC-LDPC codes with column weights of 5 and 6, very short lengths, and a girth of 8. By introducing the concept of a mirror sequence and adopting a new row-regrouping scheme, QC-LDPC codes with column weights of 7 and 8, very short lengths, and a girth of 8 are proposed for arbitrary row weights in this article via an algebraic manner under the GCD framework. Thanks to these novel algebraic methods, the lower bounds (for column weights 7 and 8) on consecutive circulant sizes are both improved by asymptotically about 20%, compared with the existing benchmarks. Furthermore, these new constructions can also offer circulant sizes asymptotically about 25% smaller than the novel bounds.
Paper Structure (15 sections, 4 figures, 3 tables)

This paper contains 15 sections, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Explicit Constructions for $J=7$
  3. Two Basic Constructions and Their Derived Ones
  4. First Group of Constructions
  5. Second Group of Constructions
  6. Lower Bound on Consecutive Circulant Sizes
  7. Circulant Size Smaller Than Lower Bound
  8. Explicit Constructions for $J=8$
  9. Two Basic Constructions and Their Derived Ones
  10. First Group of Constructions
  11. Second Group of Constructions
  12. Lower Bound on Consecutive Circulant Sizes
  13. Circulant Size Smaller Than Lower Bound
  14. Comparison with Existing Benchmarks Regarding Circulant Size
  15. Performance Comparison

Figures (4)

  • Figure 1: Circulant sizes guaranteeing girth-8 $(7,L)$-regular codes: explicitly determined value (dotted blue line) and all empirical values (marked with '+') below the lower bound on consecutive circulant sizes (dotted red line).
  • Figure 2: Circulant sizes guaranteeing girth-8 $(8,L)$-regular codes: explicitly determined value (dotted blue line) and all empirical values (marked with '+') below the lower bound on consecutive circulant sizes (dotted red line).
  • Figure 3: Performance comparison of new code (girth 8) and symmetrical code (girth 6) with circulant size of 221.
  • Figure 4: Performance comparison of new code (girth 8) and symmetrical code (girth 8) with circulant size of 559.