On nonics, ovals and codes in Desarguesian planes of even order

Summary

Abstract

A nonic in PG(2, q), q = 2^h, is defined to be either a non-degenerate conic plus its nucleus, or a degenerate conic, different from a repeated line, minus its nucleus. It is noted that every nonic is in the dual-line code of the plane, and so several questions arise. Can we have collections of nonics generating this code? Each oval (or (q + 2)-arc) is the sum (mod 2) of various numbers of nonics – what is the minimum number?

Introduction

There are four kinds of conics in a projective plane π := PG(2, q) over a finite field GF(q); see [5]. These are:

1. (1) irreducible conic, having precisely q + 1 points;

2. (2) distinct lines, having 2q + 1 points;

3. (3) lines in the quadratic extension PG(2, q²), intersecting in a point of π, and so having only one point;

4. (4) repeated line.

When q is even (and so q = 2^h) the first three types of conies define a certain point called the nucleus, which lies on the conic if and only if the conic is reducible. In the first case it is the point of intersection of all the tangents; in the second it is the intersection of the two lines; in the third case the nucleus is the only point of π that is on the conic.