NRT度量下超导数里德-所罗门码的覆盖半径与深孔
Covering Radius and Deep Holes of Hyperderivative Reed-Solomon Codes under the NRT Metric
AI总结：
本文研究NRT度量下HRS码的覆盖半径与深孔，确定其覆盖半径，通过MDS扩展等刻画深孔，关联深孔计数与超平面排列特征多项式，给出深孔构造、提出猜想并推导特殊情形的精确计数公式。
摘要：
深孔问题是编码理论中的基础问题，在码构造与密码学中有诸多重要应用，里德-所罗门码的深孔问题已受到大量关注。作为汉明度量下里德-所罗门码的推广，本文研究尼德勒特尔-罗森布鲁姆-茨法斯曼（NRT）度量下超导数里德-所罗门（HRS）码的覆盖半径与深孔。我们确定了HRS码的NRT覆盖半径，通过最大距离可分（MDS）扩展与多项式余项条件刻画其NRT深孔；还将NRT深孔的计数与关联超平面排列的特征多项式关联起来，给出三类显式NRT深孔构造，提出NRT度量下HRS码的Cheng-Murray型猜想并得到条件低层描述，利用这些刻画推导了若干特殊情形下的精确计数公式。
英文摘要：
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.
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