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.

翻译：扩展Han-Zhang码是一类线性码，其中每个码要么是非广义Reed-Solomon（非GRS）的最大距离可分（MDS）码，要么是近MDS（NMDS）码。它们在通信、密码学和存储系统中具有重要应用。虽然文献中已对扩展Han-Zhang码的许多代数性质和显式构造进行了深入研究，但其译码问题尚未得到探索。本文聚焦于基于$\ell$-纠错对（$\ell$-ECP）与深洞的译码问题。一方面，我们确定了其$\ell$-ECP的存在性及具体形式，并进一步提出基于这些$\ell$-ECP的扩展Han-Zhang码显式译码算法，该算法可在多项式时间内纠正最多$\ell$个错误，其中$\ell$约为最小距离的一半。另一方面，我们确定了扩展Han-Zhang码的覆盖半径，并刻画了两类与其最大似然译码方法密切相关的深洞。通过利用这些深洞，我们还构造了更多具有更大长度和维度的非GRS MDS码，并讨论了它们与著名的Roth-Lempel码之间的单项等价性。文中也给出了若干具体算例以支撑这些结论。