Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Francesco Pavese, Paolo Santonastaso

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.

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

Abstract

Let be the -dimensional projective space over the finite field . An arc in is a set of points with the property that any 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 -dimensional subspaces of 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 . As 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.
Paper Structure (18 sections, 21 theorems, 115 equations)

This paper contains 18 sections, 21 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Preliminaries
  4. Normal rational curve and osculating spaces
  5. Desarguesian spreads
  6. Automorphism group
  7. Quadrics
  8. New constructions of pseudo-arcs
  9. Construction
  10. The pseudo-arc Phkq is non-Desarguesian
  11. Extending the pseudo-arc Phkq
  12. Pseudo-arcs and quadrics
  13. Applications to additive MDS codes
  14. Basics on additive Hamming metric codes
  15. The geometry of additive Hamming metric codes
  16. ...and 3 more sections

Key Result

Lemma 2.4

Let $\Pi$ be a subspace of $\mathrm{PG}(hk-1,q^h)$. Then $\langle \Pi \cap \Sigma \rangle_{{\mathbb F}_{q^h}} = \langle \Pi \rangle_{{\mathbb F}_{q^h}}$ if and only if $\Pi^{\Psi} = \Pi$.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: see lunardon1999normal
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 32 more