Representative Ordered Statistics Decoding of Staircase Matrix Codes

We propose a class of codes, referred to as staircase matrix codes (SMCs), which have staircase-like generator matrices or parity-check matrices. We illustrate that polar codes and Reed-Muller (RM) codes are (equivalent to) two instances of SMCs. The most distinguished feature of the SMCs is that the staircase-like matrices enable parallel implementation of the Gaussian elimination (GE). Then we propose a representative ordered statistics decoding (ROSD) algorithm for the SMCs. Different from the conventional OSD, which forms the most reliable basis (MRB) by selecting reliable bits globally, the proposed ROSD forms an extended basis by selecting relatively reliable bits locally at least one from each staircase. We demonstrate by simulation that the ROSD has a similar performance to the OSD with local constraints (LC-OSD) and that the proposed random SMCs can outperform the RM codes and the polar codes. To further reduce the decoding delay and improve the performance, we propose a heuristic construction of staircase generator matrix codes (SGMCs) and analyze the ensemble weight spectrum (related to performance) and the quality of MRB (related to average number of test error patterns (TEPs)) for the heuristic construction. The simulation results show that, the proposed heuristic construction can reduce the average number of TEPs of the ROSD and provide a potential reduction in average decoding delay in the high signal-to-noise ratio (SNR) region. Furthermore, the proposed heuristic construction can approach the RCU bounds in a wide range of code rates.