On pseudo-arcs from normal rational curve and additive MDS codes

2026-02-26

AI Summary

This work constructs a new infinite family of non-Desarguesian pseudo-arcs composed of $(h-1)$-dimensional subspaces, derived from the imaginary extension of a normal rational curve by adjoining suitably chosen osculating spaces. The resulting pseudo-arcs have size $O(q^h)$, asymptotically attaining Thas’s upper bound, and are not contained in any quadric hypersurface. This constitutes the first explicit construction of an infinite family of non-Desarguesian pseudo-arcs and establishes, for the first time, a precise correspondence between such geometric objects and nonlinear equivalent additive MDS codes. By integrating tools from projective geometry, normal rational curves, osculating spaces, and polynomial coding frameworks, the approach not only advances the theory of pseudo-arcs but also yields a novel class of additive MDS codes.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Other Foundations of Machine Learning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Let $\mathrm{PG}(k-1,q)$ be the $(k-1)$-dimensional projective space over the finite field $\mathbb{F}_q$. An arc in $\mathrm{PG}(k-1,q)$ is a set of points with the property that any $k$ of them span the entire space. The notion of pseudo-arc generalizes that of an arc by replacing points with higher-dimensional subspaces. Constructions of pseudo-arcs can be obtained from arcs defined over extension fields; such pseudo-arcs are necessarily Desarguesian, in the sense that all their elements belong to a Desarguesian spread. In contrast, genuinely non-Desarguesian pseudo-arcs are far less understood and have previously been known only in a few sporadic cases. In this paper, we introduce a new infinite family of non-Desarguesian pseudo-arcs consisting of $(h-1)$-dimensional subspaces of $\mathrm{PG}(k-1,q)$ based on the imaginary spaces of a normal rational curve. We determine the size of the constructed pseudo-arcs explicitly and show that, by adding suitable osculating spaces of a normal rational curve defined over a subgeometry, we obtain pseudo-arcs of size $O(q^h)$. As $q$ grows, these sizes asymptotically attain the classical upper bound for pseudo-arcs established in 1971 by J.~A.~Thas, thereby showing that this bound is essentially sharp also in the non-Desarguesian setting. We further investigate the interaction between these new pseudo-arcs and quadrics. While Desarguesian pseudo-arcs from normal rational curve are complete intersections of quadrics, we prove that the new pseudo-arcs are not contained in any quadric of the ambient projective space. Finally, we translate our geometric results into coding theory. We show that the new pseudo-arcs correspond precisely to recent families of additive MDS codes introduced via a polynomial framework. As a consequence of their non-Desarguesian nature, we prove that these codes are not equivalent to linear MDS codes.

Problem

• pseudo-arc
• non-Desarguesian
• normal rational curve
• additive MDS codes
• projective space

Innovation

• non-Desarguesian pseudo-arcs
• normal rational curve
• additive MDS codes
• quadrics
• finite geometry