Lifted Gabidulin Construction for LDPC Representations of Finite Geometry Codes

2026-06-09

AI Summary

This work addresses the poor iterative decoding performance of low-density parity-check (LDPC) codes derived from finite geometries, which stems from the dense structure and abundance of short cycles in their conventional parity-check matrices. To overcome this limitation, the authors propose a sparsification method based on pencil selection, formulated as a constant-dimension subspace packing problem. They introduce lifted Gabidulin codes to explicitly construct sparse parity-check matrices tailored for both affine and projective geometries—preserving underlying algebraic structures while effectively eliminating short cycles. The approach successfully yields sparse matrices of length up to 1024. Simulation results demonstrate a coding gain of approximately 0.5 dB over 5G LDPC codes at a block error rate of $10^{-7}$, with no evident error floor observed.

Abstract

Finite geometry (FG) codes combine the algebraic properties of classical block codes with the iterative belief propagation (BP) decoding ability of low-density parity-check~(LDPC) codes. However, exploiting both advantages in practice is hindered by the fact that the standard incidence matrix between $(μ+1)$-flats and points is dense and contains many short cycles for any flat dimension $μ\geq 1$. In this work, we propose to sparsify the decoding matrix based on pencil selection, formulated as a constant-dimension subspace packing problem and solved explicitly using lifted Gabidulin codes. For both affine and projective geometries, sparse parity-check matrices are constructed and verified for FG codes of lengths up to $1024$. Simulations on four FG codes show no visible error floor and around $0.5$~dB gain over corresponding 5G LDPC codes at a block error rate of $10^{-7}$.

Problem

• finite geometry codes
• dense incidence matrix
• short cycles
• LDPC decoding
• parity-check matrix

Innovation

• lifted Gabidulin codes
• finite geometry codes
• sparse parity-check matrix
• subspace packing
• LDPC decoding