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Practical implementation of geometric quasi-cyclic LDPC codes

Simeon Ball, Tomàs Ortega

TL;DR

This work develops explicit, highly structured QC representations for LDPC codes derived from classical finite geometries, enabling long codes up to length $n\approx 4\cdot 10^{5}$ with practical encoding and decoding. By constructing QC check and generator matrices $H$ and $G$ from incidences in generalized quadrangles and projective/affine spaces, and by removing spreads to increase block size, the authors produce implementable subcodes $C'$ that retain QC structure and near-capacity performance. Performance results in AWGN show a near-1 dB gap to capacity with no visible error floors for several $Q(5,q)$ codes, and some codes outperform DVB-S2 at comparable rates and lengths, highlighting the practical impact for long, efficient LDPC implementations. The paper provides a systematic, geometry-based framework for QC-LDPC code construction with explicit matrices, enabling scalable, high-rate, long-block-length codes suitable for communications and storage applications, along with extensive guidance for future extensions to additional geometries and incidences.

Abstract

We detail for the first time a complete explicit description of the quasi-cyclic structure of all classical finite generalized quadrangles. Using these descriptions we construct families of quasi-cyclic LDPC codes derived from the point-line incidence matrix of the quadrangles by explicitly calculating quasi-cyclic generator and parity check matrices for these codes. This allows us to construct parity check and generator matrices of all such codes of length up to 400000. These codes cover a wide range of transmission rates, are easy and fast to implement and perform close to Shannon's limit with no visible error floors. We also include some performance data for these codes. Furthermore, we include a complete explicit description of the quasi-cyclic structure of the point-line and point-hyperplane incidences of the finite projective and affine spaces.

Practical implementation of geometric quasi-cyclic LDPC codes

TL;DR

This work develops explicit, highly structured QC representations for LDPC codes derived from classical finite geometries, enabling long codes up to length with practical encoding and decoding. By constructing QC check and generator matrices and from incidences in generalized quadrangles and projective/affine spaces, and by removing spreads to increase block size, the authors produce implementable subcodes that retain QC structure and near-capacity performance. Performance results in AWGN show a near-1 dB gap to capacity with no visible error floors for several codes, and some codes outperform DVB-S2 at comparable rates and lengths, highlighting the practical impact for long, efficient LDPC implementations. The paper provides a systematic, geometry-based framework for QC-LDPC code construction with explicit matrices, enabling scalable, high-rate, long-block-length codes suitable for communications and storage applications, along with extensive guidance for future extensions to additional geometries and incidences.

Abstract

We detail for the first time a complete explicit description of the quasi-cyclic structure of all classical finite generalized quadrangles. Using these descriptions we construct families of quasi-cyclic LDPC codes derived from the point-line incidence matrix of the quadrangles by explicitly calculating quasi-cyclic generator and parity check matrices for these codes. This allows us to construct parity check and generator matrices of all such codes of length up to 400000. These codes cover a wide range of transmission rates, are easy and fast to implement and perform close to Shannon's limit with no visible error floors. We also include some performance data for these codes. Furthermore, we include a complete explicit description of the quasi-cyclic structure of the point-line and point-hyperplane incidences of the finite projective and affine spaces.
Paper Structure (13 sections, 13 theorems, 127 equations, 6 figures, 7 tables)

This paper contains 13 sections, 13 theorems, 127 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Quasi cyclic generator and check matrices
  3. Quasi-cyclic LDPC codes from classical geometries.
  4. The quasi-cyclic representations of the classical finite generalized quadrangles.
  5. The elliptic quadric quadrangle Q(5,q)
  6. The symplectic quadrangle W(3,q).
  7. The Hermitian quadrangle H(4,q2)
  8. The quasi-cyclic representation of the projective and affine spaces
  9. The projective space PG(k-1,q).
  10. The affine space AG(k,q).
  11. The parameters of the classical GQ LDPC codes of length up to 400,000.
  12. Performance of classical GQ LDPC codes
  13. Further work

Key Result

Theorem 1

The quasi-cyclic representation of $Q(5,q) \setminus \Sigma_C$ is given by $i \in \mathrm{H}^{\mathrm{rep}}_{x,a}$ for $x \in P_1$ and $a \in L_1$ if and only if where $i \in \{0,\ldots,b-1\}$, and $b=q^3+1$.

Figures (6)

  • Figure 1: The quasi-cyclic check matrix H for Q$(5,3)$. Dotted elements correspond to ones, and the rest are zeros.
  • Figure 2: The matrix P$^{\mathrm{rep}}$ (top) and corresponding quasi-cyclic generator matrix G for Q$(5,3)$ (bottom). Dotted elements correspond to ones, and the rest are zeros.
  • Figure 3: The performance of the LDPC codes from Q$(5,7)$, Q$(5,9)$, and Q$(5,13)$. The corresponding PPV bounds are plotted in dashed lines. Each code is decoded using 25 SPA iterations.
  • Figure 4: The performance of the LDPC codes from Q$(5,7)$, Q$(5,9)$, and (16200, 14400) DVB-S2. Each code is decoded using 50 SPA iterations.
  • Figure 5: Q$(5,9)$ code performance compared with a (16200, 14400) DVB-S2 code of the same rate, decoded with different number of SPA iterations.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • ...and 16 more