On Generalized Reed-Solomon Codes

In this paper, we prove that the generalized Reed-Solomon (RS) codes are capacity-achieving over binary-input output-symmetric (BIOS) channels, in terms of frame error rate (FER) under maximum likelihood (ML) decoding. In the finite-length region, we present the ordered statistics decoding with local constraints (LC-OSD) algorithm for the generalized RS codes. In particular, the extended most reliable basis (MRB) is derived based on a systematic matrix calculated by the parallel Lagrange interpolation, accelerating the conventional Gaussian elimination (GE). Additionally, we propose a joint source-channel coding (JSCC) scheme that incorporates generalized RS codes and classified enumerative (CE) coding, where the partition of the source is optimized by the k-means++ clustering algorithm. At the transmitter, we implement the CE coding to encode the source information. Then the variable-length codeword of the CE coding is transformed into a fixed-length codeword by the multiple-rate generalized RS encoding and superimposed with a class label for transmission. At the receiver, parallel LC-OSD is performed to recover the source. Simulation results demonstrate that the proposed JSCC scheme outperforms the double polar JSCC scheme (exhibiting a coding gain of up to 1.4 dB) and the JSCC scheme based on polarizing matrix extension (exhibiting a coding gain of up to 0.6 dB), as predicted by Gallager’s JSCC bound.