Biplanes

This page lists all known biplanes, that is symmetric 2-(v,k,2) designs. The list is known to be complete only for values of k up to 9.

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Table of Contents

• Introduction
• Known biplanes
• Notes on the biplanes

Introduction

A symmetric 2-(v,k,1) design is called a projective plane . The well-known construction based on finite fields shows that there are infinitely many projective planes; all known projective planes have their order k-1 being a prime power.

For any other fixed value x the question of whether there are an infinite number of symmetric 2-(v,k,x) designs is unresolved. In particular, when x=2, such a design is called a biplane and there are only a small number of examples known. Resolving the question of whether there are an infinite number of biplanes would be an extremely significant result in design theory.

This catalogue lists all the known biplanes, and gives some simple data regarding each one. The catalogue is known to be complete for k=9 and below, but the full number of biplanes on 56 and 79 points is currently unknown. For even more ambitious researchers, the next open cases are 2-(121,16,2) designs and 2-(154,18,2).

Known biplanes

The following table lists the known biplanes. Each biplane is given explicitly as a list of blocks preceded by the parameters of the design (in the t-(v,k,l) format).

The columns Aut. group and Orbit partition refer to the automorphism group of the bipartite point/block incidence graph of the design. In particular if the design is self-dual then there is an automorphism exchanging points and blocks, and the group given will be twice as large as you may have expected. If the design is not self-dual then the orbit partition lists the point orbits first, followed by the symbol --, followed by the block orbits.

The column labelled p-rank refers to the rank of the point/block incidence matrix over the unique prime p that divides the order of the design.

Notes

For the moment the notes are limited merely to the Hussain chain structure of each biplane. For more details about Hussain chains see most design theory books (e.g. Hall, Hughes and Piper, Beth, Jungnickel and Lenz).

Biplanes with k=3

The Hussain chain structures are as follows:

Biplanes with k=4

The Hussain chain structures are as follows:

Biplanes with k=5

The Hussain chain structures are as follows:

Biplanes with k=6

The Hussain chain structures are as follows:

Biplanes with k=9

The Hussain chain structures are as follows:

Biplanes with k=11

The Hussain chain structures are as follows:

Biplanes with k=13

The Hussain chain structures are as follows: