Unique Decoding of Extended Subcodes of GRS Codes Using Error-Correcting Pairs
Yang Li, Zhenliang Lu, San Ling, Shixin Zhu, Kwok Yan Lam
TL;DR
This work addresses unique decoding of extended subcodes of GRS (ESGRS) codes, a class where each code is either non-GRS MDS or NMDS. It develops and analyzes $\ell$-error-correcting pairs ($\ell$-ECPs) for ESGRS codes, providing explicit $\ell$-ECP constructions and, for MDS and NMDS cases, characterizing when such pairs exist. Leveraging these ECPs, the authors present a concrete decoding algorithm with time complexity $O(n^3)$ that can uniquely correct up to $\ell$ errors (where $\ell=\left\lfloor\frac{d(C_k)}{2}\right\rfloor$ or as specified by the code) and discuss potential reductions to $O(qn)$ using alternative methods. The paper also determines the covering radius and deep holes of ESGRS codes, uses deep holes to construct more non-GRS MDS codes, and clarifies connections to Roth–Lempel codes. Overall, the results offer a practical, algebraic approach to decoding ESGRS codes and expand the toolkit for designing robust MDS/NMDS codes for storage and cryptography.
Abstract
Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication, cryptography, and storage systems. While many algebraic properties and explicit constructions of extended Han-Zhang codes have been well studied in the literature, their decoding has been unexplored. In this paper, we focus on their decoding problems in terms of $\ell$-error-correcting pairs ($\ell$-ECPs) and deep holes. On the one hand, we determine the existence and specific forms of their $\ell$-ECPs, and further present an explicit decoding algorithm for extended Han-Zhang codes based on these $\ell$-ECPs, which can correct up to $\ell$ errors in polynomial time, with $\ell$ about half of the minimum distance. On the other hand, we determine the covering radius of extended Han-Zhang codes and characterize two classes of their deep holes, which are closely related to the maximum-likelihood decoding method. By employing these deep holes, we also construct more non-GRS MDS codes with larger lengths and dimensions, and discuss the monomial equivalence between them and the well-known Roth-Lempel codes. Some concrete examples are also given to support these results.