Coding Theorem for Generalized Reed-Solomon Codes

In this paper, we prove that the sub-field images of generalized Reed-Solomon (RS) codes can achieve the symmetric capacity of p-ary memoryless channels. Unlike the totally random linear code ensemble, as a class of maximum distance separable (MDS) codes, the generalized RS code ensemble lacks the pair-wise independence among codewords and has non-identical distributions of nonzero codewords. To prove the coding theorem for the p-ary images of generalized RS codes, we analyze the exponential upper bound on the error probability of the generalized RS code in terms of its spectrum using random coding techniques. In the finite-length region, we present an ML decoding algorithm for the generalized RS codes over the binary erasure channels (BECs). In particular, the algebraic structure of the generalized RS codes allows us to implement the parallel Lagrange interpolation to derive an ordered systematic matrix. Subsequently, we can reconstruct the ML codeword through a change of basis, accelerating the conventional Gaussian elimination (GE), as validated in the simulation results. Additionally, we apply this decoding technique to the LC-OSD algorithm over the additive white Gaussian noise (AWGN) channels with binary phase shift keying (BPSK) modulation and three-level pulse amplitude modulation (3PAM). Simulation results show that, in the high-rate region, generalized RS codes defined over fields of characteristic three with 3-PAM perform better than those defined over fields of characteristic two with BPSK.