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

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.