Invariants of Quadratic Forms and applications in Design Theory

TL;DR

This paper demonstrates that linear algebraic methods, similar to Jordan Canonical Form, can be used to analyze quadratic forms and derive non-existence results in design theory, simplifying previous approaches.

Contribution

It introduces a purely linear-algebraic approach to study invariants of quadratic forms and applies it to problems in design theory, including incidence matrix decomposition.

Findings

Linear algebraic methods are sufficient for analyzing quadratic form invariants.

Simplified proofs of non-existence results in design theory.

New application to decomposing incidence matrices of symmetric designs.

Abstract

The study of regular incidence structures such as projective planes and symmetric block designs is a well established topic in discrete mathematics. Work of Bruck, Ryser and Chowla in the mid-twentieth century applied the Hasse-Minkowski local-global theory for quadratic forms to derive non-existence results for certain design parameters. Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic in nature and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs.