Covering Radius and Deep Holes of Hyperderivative Reed-Solomon Codes under the NRT Metric

Abstract

The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes under Hamming metric, we study the covering radius and deep holes of Hyperderivative Reed-Solomon (HRS) codes under the Niederreiter-Rosenbloom-Tsfasman (NRT) metric. We determine the NRT covering radius of HRS code and characterize its NRT deep holes via Maximum Distance Separable (MDS) extensions and polynomial remainder conditions. We further relate the enumeration of NRT deep holes to the characteristic polynomial of an associated hyperplane arrangement. We also provide three classes of explicit NRT deep-hole constructions, propose a Cheng-Murray type conjecture for HRS codes under the NRT metric and obtain a conditional low-layer description. Using these characterizations, we derive exact enumeration formulas in several special cases.