Lifted Gabidulin Construction for LDPC Representations of Finite Geometry Codes
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.