Binary Images of Generalized Reed-Solomon Codes are Capacity Achieving

In this paper, we prove that the binary images of generalized Reed-Solomon (RS) codes are capacity-achieving over binary-input output-symmetric channels, in terms of frame error rate (FER) under maximum likelihood (ML) decoding. In the finite-length region, we propose to decode the generalized RS codes using the ordered statistics decoding with local constraints (LC-OSD) algorithm. Unlike LC-OSD of other codes, we replace Gaussian elimination with the parallel Lagrange interpolation plus change-of-basis to reduce the decoding latency. Simulation results show that the generalized RS codes can approach the random coding union bounds in a wide range of code rates.