A $q$-analogue of the rational normal curve and linearized Reed-Solomon codes

2026-06-11

AI Summary

This work addresses the poorly understood geometric structure of linearized Reed–Solomon codes under the sum-rank metric, for which effective distinguishing tools have been lacking. The authors introduce, for the first time, a $q$-analogue of the rational normal curve and establish its connection to algebraic varieties, revealing that the corresponding point sets satisfy numerous hypersurface equations of degree $q+1$. They further determine the Hilbert function and Castelnuovo–Mumford regularity of the associated coordinate ring. Additionally, the classical Schur product technique is extended to the sum-rank metric setting. These contributions yield a refined geometric characterization of linearized Reed–Solomon codes for specific parameters, deepen the understanding of Gabidulin codes, and offer a novel algebraic-geometric perspective for distinguishing sum-rank metric codes.

Abstract

The relationship between linear codes in the Hamming metric and projective algebraic varieties has led to deep interactions between coding theory and algebraic geometry, with classical examples such as Reed-Solomon codes and the rational normal curve. On the other hand, the sum-rank metric has recently gained attention due to applications in network coding, distributed storage, and post-quantum cryptography, with linearized Reed-Solomon codes emerging as optimal constructions. Despite recent advances, their structural and geometric properties are still not fully understood, and existing distinguishers remain limited. In this paper, we develop a geometric framework for linearized Reed-Solomon codes by considering a $q$-analogue of the rational normal curve. This yields a geometric characterization for certain parameter choices and reveals that the corresponding sets of points satisfy unexpectedly many $(q+1)$-degree hypersurface conditions. Our approach extends Schur-product-based techniques from the Hamming and rank-metric settings to the sum-rank metric case. Finally, we study the Hilbert function of the associated coordinate ring, providing a detailed description of its behavior and identifying its regularity, which also sheds new light on Gabidulin codes.

Problem

• linearized Reed-Solomon codes
• sum-rank metric
• geometric characterization
• q-analogue
• rational normal curve

Innovation

• q-analogue
• linearized Reed-Solomon codes
• sum-rank metric
• rational normal curve
• Hilbert function