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

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.