More nonexistence results for symmetric pair coverings

A $(v,k,λ)$-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 $λ$ 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 $r_{xy}-λ$, 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.

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