Unique Decoding of Extended Subcodes of GRS Codes Using Error-Correcting Pairs

TL;DR

This paper introduces a polynomial-time decoding algorithm for extended Han-Zhang codes using error-correcting pairs, characterizes deep holes, and constructs new MDS codes, advancing decoding theory for these codes.

Contribution

It provides the first explicit decoding method for extended Han-Zhang codes based on $ extit{ extbf{ extlangle} extell extlangle} $-error-correcting pairs, and characterizes their deep holes and covering radius.

Findings

Decoding algorithm corrects up to half the minimum distance.

Determined the covering radius of extended Han-Zhang codes.

Constructed new non-GRS MDS codes related to Roth-Lempel codes.

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.