Optimized design of quasi-GRS codes with multi-step simplified MLD

Abstract

In this manuscript, inspired by the special construction of generalized Reed-Solomon (GRS) codes, we prove that any linear block code can be contained by a high-dimensional GRS code from the perspective of the isomorphism between the code space and the polynomial space. Then, a new code family, named quasi-GRS codes, is constructed, which can exactly contain a GRS code of lower 1-dimensional (1-D) and be contained by a higher 1-D GRS code. Furthermore, the exhaustive search approach is developed to assist in selecting the quasi-GRS code with the optimal code weight distribution. In addition, a multi-step simplified maximum likelihood decoding (MSS-MLD) is proposed to efficiently decode the quasi-GRS coding scheme, whose decoding complexity is significantly reduced at an acceptable performance loss compared to the optimal MLD. From the Monte-Carlo simulation, the optimized quasi-GRS coding scheme outperforms the existing non-binary Bose-Chaudhuri-Hocquenghem (BCH) coding scheme with a similar code rate by a margin of over 1 dB at the high signal-to-noise (SNR) region under identical channel conditions.