Improved Decoding Algorithms for MDS and Almost-MDS Codesfrom Twisted GRS Codes
Guodong Wang, Hongwei Liu, Jinquan Luo
TL;DR
The paper addresses decoding of twisted generalized Reed-Solomon (TGRS) codes, including almost-MDS variants, by providing a precise key-equation characterization and optimized decoding algorithms based on the extended Euclidean algorithm. It develops a syndrome-based framework and proves uniqueness of the decoding pair (sigma, tau) up to a scalar, while reducing the search space for practical decoding. The authors extend these methods to twisted Goppa codes as subfield subcodes and demonstrate improved performance over prior work. Overall, the work broadens efficient decoding capabilities for TGRS and twisted Goppa codes with significant potential impact on communications and cryptography.
Abstract
In this paper, firstly, we study decoding of a general class of twisted generalized Reed-Solomon (TGRS) codes and provide a precise characterization of the key equation for TGRS codes and propose a decoding algorithm. Secondly, we further study decoding of almost-MDS TGRS codes and provide a decoding algorithm. These two decoding algorithms are more efficient in terms of performance compared with the decoding algorithms presented in [Sun et al., IEEE-TIT, 2024] and [Sui et al., IEEE-TIT, 2023] respectively.