✦ LIBER ✦

Sets with few Intersection Numbers from Singer Subgroup Orbits

✍ Scribed by J. Coykendall; J. Dover

Book ID: 102570042
Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 116 KB
Volume: 22
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.2000.0471

No coin nor oath required. For personal study only.

✦ Synopsis

Using a Singer cycle in Desarguesian planes of order q ≡ 1 (mod 3), q a prime power, Brouwer [2] gave a construction of sets such that every line of the plane meets them in one of three possible intersection sizes. These intersection sizes x, y, and z satisfy the system of equations

Brouwer claimed that this system has a unique solution in integers. Further, Brouwer noted that for q a perfect square, this system has a solution for which two of the variables are equal, ostensibly implying that when q is a square the constructed set has only two intersection numbers.

In this paper, we perform a detailed analysis which shows that this system does not in general have a unique solution. In particular, the constructed sets when q is a square might have three intersection numbers. The cases for which this occurs are completely determined.

📜 SIMILAR VOLUMES

Sets in a finite plane with few intersec

Sets in a finite plane with few intersection numbers and a distinguished point

✍ J.W.P. Hirschfeld; T. Szőnyi 📂 Article 📅 1991 🏛 Elsevier Science 🌐 English ⚖ 742 KB

There have been many characterizations of the classical curves in PG(2, q) given by the zeros of quadratic and Hermitian forms. The next step is to characterize pencils of such curves. Here it is done in the case that the pencils have a single base point. A key result that emerges from the investiga