The properties of bordered matrix of symmetric block design

TL;DR

This paper introduces the bordered matrix concept for symmetric block designs, providing new necessary conditions that help determine the non-existence of certain finite projective planes, especially for open cases like n=10 and 12.

Contribution

The author proposes the bordered matrix as a novel tool to derive necessary conditions, extending the ability to exclude the existence of symmetric designs beyond existing theorems.

Findings

Excluded the existence of projective planes for n=10, 12, 15, 18, 20, 24, 26, 28

Provided a new method applicable to large n cases

Identified symmetric designs that cannot exist using the new conditions

Abstract

An incidence structure consists simply of a set P of points and a set B of blocks, with a relation of incidence between points and blocks.A symmetric (v,k,\lambda) block design is the subject of this paper. The symmetric (n^2+n+1, n+1,1) block design is a projective plane of order n. Despite much research no one has uncovered any further necessary conditions for the existence of a symmetric (v,k,\lambda) design apart from the equation (v-1)\lambda=k(k-1), Schutzenberger's Theorem and the Bruck-Ryser-Chowla Theorem. For no (v,k,\lambda) satisfying these requirements has it been shown that a symmetric (v,k,\lambda) design does not exist. Projective planes of order n exist for all prime powers n (aside from PG(2,n) a host of other constructions are known ) but for no other n is a construction known. The first open values are n=10, 12, 15, 18, 20, 24, 26$ and 28. It was proved by a computer search that there does not exist any projective plane of order 10 by Lam, C.W.H., Thiel, L. and Swiercz, S. Whether there exists any projective plane of order 12 is still open.The author introduces the bordered matrix of a (v,k,\lambda) symmetric design and gives some new necessary conditions for the existence of the symmetric (v,k,\lambda) design. As their application it is easy to determine that there does not exist finite projective plane of order n if n is one of the first open values 10, 12, 15, 18, 20, 24, 26 and 28, for which the Bruck-Ryser-Chowla Theorem can not be used. For large n the new method is also valid. Also some symmetric designs are excluded by the new method.