AI 中文总结
研究超导数里德-所罗门码对偶性问题，利用留数理论得出其欧几里得对偶的显式表示，发现其与反向 HRS 评估码相关，基于此研究反向自对偶 HRS 码，建立准则并构造多个码族。
AI 中文摘要
超导数里德-所罗门（HRS）码在 Niederreiter-Rosenbloom-Tsfasman 度量下是一类最大距离可分码，可视为经典里德-所罗门码的导数评估扩展。本文研究 HRS 码的对偶性问题，利用留数理论推导其欧几里得对偶的显式分量表示，发现其一般不是 HRS 码，而是与反向 HRS 评估码块上三角等价。对于低重全域 HRS 码，对偶是 HRS 码的行反转。基于此研究反向自对偶 HRS 码，建立准则并从加性和乘性陪集结构构造了几个码族。
英文摘要
Hyperderivative Reed-Solomon (HRS) codes form a class of maximum-distance-separable codes under the Niederreiter-Rosenbloom-Tsfasman metric and may be viewed as a derivative-evaluation extension of classical Reed-Solomon codes. For generalized Reed-Solomon codes, the Euclidean dual is again a generalized Reed-Solomon code. In this paper, we investigate the corresponding duality problem for HRS codes. Using a residue-theoretic argument, we derive an explicit component-wise representation for the Euclidean dual of an HRS code. The formula shows that, in general, the Euclidean dual is not an HRS code. Instead, it is blockwise upper-triangularly equivalent to a reverse-order HRS evaluation code, where the reversal occurs in the hyperderivative orders within each evaluation block. In particular, for full-domain HRS codes with low multiplicity, the triangular transformations reduce to diagonal scalings, and the Euclidean dual is obtained as the row reversal of an HRS code. Based on this reverse-order dual structure, we further study reverse self-dual HRS codes. We establish explicit criteria for reverse self-duality and construct several families from additive and multiplicative coset structures.
