More nonexistence results for symmetric pair coverings

TL;DR

This paper extends nonexistence results for symmetric pair coverings by adapting matrix rational congruence methods, proving certain cyclic and excess-specific symmetric coverings cannot exist.

Contribution

It introduces new nonexistence proofs for symmetric coverings using rational matrix congruence techniques, expanding prior determinant-based methods.

Findings

Certain cyclic symmetric coverings are proven not to exist.

Nonexistence of symmetric coverings with specific excesses is established.

The methods generalize previous nonexistence results for symmetric pair coverings.

Abstract

A $(v,k,\lambda)$-covering is a pair $(V, \mathcal{B})$, where $V$ is a $v$-set of points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ (called blocks), such that every unordered pair of points in $V$ is contained in at least $\lambda$ blocks in $\mathcal{B}$. The excess of such a covering is the multigraph on vertex set $V$ in which the edge between vertices $x$ and $y$ has multiplicity $r_{xy}-\lambda$, where $r_{xy}$ is the number of blocks which contain the pair $\{x,y\}$. A covering is symmetric if it has the same number of blocks as points. Bryant et al.(2011) adapted the determinant related arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the nonexistence of certain symmetric coverings with $2$-regular excesses. Here, we adapt the arguments related to rational congruence of matrices and show that they imply the nonexistence of some cyclic symmetric coverings and of various symmetric coverings with specified excesses.