Practical implementation of geometric quasi-cyclic LDPC codes

TL;DR

This paper provides a detailed construction of quasi-cyclic LDPC codes from finite geometries, enabling efficient implementation and near-capacity performance for large code lengths.

Contribution

It offers the first explicit description of the quasi-cyclic structure of classical finite generalized quadrangles and constructs large LDPC codes with explicit matrices.

Findings

Codes perform close to Shannon's limit

Constructed codes of length up to 400,000

No visible error floors in performance

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.