Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Giuseppe Veronese and Ernst Witt -- Neighbours in PG(5,3)

Hans Havlicek

Abstract

Let $P$ be a point of the Veronese surface $\Vcal$ in \PG53. Then thereare four conics of $\Vcal$ through $P$. We show that the internal points of those conics form a 12-cap which is a point model for Witt's 5-$(12,6,1)$ design. In fact, this construction is "dual" to a similar construction that has been established by the author. We give an explicit parametrization of the cap $\Kcal$; the domain is a dual affine plane which arises from \PG23 by removing one point. Thus, as a by--product, we obtain an easy approach to the extended ternary Golay code $G_{12}$. Finally, we discuss some other procedures that yield 12-sets of points from the Veronese surface.

Giuseppe Veronese and Ernst Witt -- Neighbours in PG(5,3)

Abstract

Let be a point of the Veronese surface in \PG53. Then thereare four conics of through . We show that the internal points of those conics form a 12-cap which is a point model for Witt's 5- design. In fact, this construction is "dual" to a similar construction that has been established by the author. We give an explicit parametrization of the cap ; the domain is a dual affine plane which arises from \PG23 by removing one point. Thus, as a by--product, we obtain an easy approach to the extended ternary Golay code . Finally, we discuss some other procedures that yield 12-sets of points from the Veronese surface.

Paper Structure

This paper contains 4 sections, 2 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Veronese Surface and its Dual
  3. Point Models of $W_{12}$
  4. Replacing Conics of the Veronese Surface

Key Result

Theorem 1

Let $P$ be a point of the Veronese surface ${\cal V}$ in $\hbox{\rm PG}(5,3)$. The four conics of ${\cal V}$ through $P$ are denoted by $m_k$ ($k\in F\cup\{\infty\}$). The set of internal points of each $m_k$ is written as $\Delta_k$. Also let $c^\ast$ be the set of osculating primes of ${\cal V}$ t

Figures (1)

  • Figure 1: A conic in $\hbox{\rm PG}(2,3)$

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2