  In this paper, we propose a new linear-equation ordered-statistics decoding
(LE-OSD). Unlike the OSD, LE-OSD uses high reliable parity bits rather than
information bits to recover the codeword estimates, which is equivalent to
solving a system of linear equations (SLE). Only test error patterns (TEPs)
that create feasible SLEs, referred to as the valid TEPs, are used to obtain
different codeword estimates. We introduce several constraints on the Hamming
weight of TEPs to limit the overall decoding complexity. Furthermore, we
analyze the block error rate (BLER) and the computational complexity of the
proposed approach. It is shown that LE-OSD has a similar performance as OSD in
terms of BLER, which can asymptotically approach Maximum-likelihood (ML)
performance with proper parameter selections. Simulation results demonstrate
that the LE-OSD has a significantly reduced complexity compared to OSD,
especially for low-rate codes, that usually require high decoding order in OSD.
Nevertheless, the complexity reduction can also be observed for high-rate
codes. In addition, we further improve LE-OSD by applying the decoding stopping
condition and the TEP discarding condition. As shown by simulations, the
improved LE-OSD has a considerably reduced complexity while maintaining the
BLER performance, compared to the latest OSD approach from literature.
