Duality and Reverse Self-Dual Constructions for Hyperderivative Reed-Solomon Codes

2026-07-22

AI Summary

This study investigates the structural properties and self-dual constructions of hyperbolic Reed–Solomon (HRS) codes under Euclidean duality. By employing residue theory from complex analysis, the authors provide, for the first time, an explicit component-wise expression for the dual code and establish a block upper-triangular equivalence between the dual code and the reversed-order hyperbolic code. It is shown that, in general, the dual of an HRS code is not itself an HRS code; however, in the low-multiplicity full-domain setting, the dual simplifies to a combination of diagonal scaling and row reversal. Building on this insight, the paper introduces the novel notion of “reversed-order self-dual” HRS codes, derives necessary and sufficient conditions for their existence, and constructs several new families of such codes, thereby enriching both the theoretical framework and concrete examples of self-dual codes in algebraic coding theory.

Abstract

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.

Problem

• Hyperderivative Reed-Solomon codes
• Euclidean dual
• reverse self-dual
• duality
• Niederreiter-Rosenbloom-Tsfasman metric

Innovation

• Hyperderivative Reed-Solomon codes
• Euclidean dual
• reverse self-dual
• residue-theoretic argument
• Niederreiter-Rosenbloom-Tsfasman metric