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

TL;DR

This paper introduces a new infinite family of non-Desarguesian pseudo-arcs derived from normal rational curves, analyzes their size and geometric properties, and connects them to novel additive MDS codes that are not linearly equivalent.

Contribution

The paper constructs a new class of non-Desarguesian pseudo-arcs from normal rational curves and links them to additive MDS codes, expanding understanding of pseudo-arc structures and code equivalence.

Findings

Constructed pseudo-arcs of size asymptotically reaching classical bounds

Proved new pseudo-arcs are not contained in any quadric

Established correspondence between pseudo-arcs and additive MDS codes

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.