Random Staircase Generator Matrix Codes

In this paper, we propose a class of codes, referred to as random staircase generator matrix codes (SGMCs), which have staircase-like generator matrices. In the infinite-length region, we prove that the random SGMC is capacity-achieving over binary-input output-symmetric (BIOS) channels. In the finite-length region, we present the representative ordered statistics decoding with local constraints (LC-ROSD) for the SGMCs. The most distinguished feature of the SGMCs with LC-ROSD is that the staircase-like matrices enable parallel implementation of the Gaussian elimination (GE), avoiding the serial GE of conventional OSD and supporting a potential low decoding latency, as implied from simulations. To analyze the performance of random SGMCs in the finite-length region, we derive the ensemble weight spectrum and invoke the conventional union bound. We also derive a partially random coding union (RCU) bound, which is tighter than the conventional one and can be used as a criterion to design the SGMCs. The numerical results show that the random SGMCs with the LC-ROSD exhibit a significant decoding time improvement compared to that with the conventional OSD. They also show that the tailored SGMCs with the LC-ROSD can approach the finite-length performance bound, outperforming the 5G low-density parity-check (LDPC) codes, 5G polar codes and Reed-Muller (RM) codes.