What theorems may come From clues in finite geometry to a construction of a CA$(3q^4-2; 4, \frac{q^2+1}{2}, q)$

Kianoosh Shokri (University of Ottawa)

A \emph{strength-\(t\)} \emph{covering array}, denoted by CA\((N; t, k, v)\), is an \(N \times k\) array over a \(v\)-set such that in any \(t\)-set of columns, each \(t\)-tuple occurs at least once in a row.

Almost every theorem in mathematics has interesting stories of success and failure in the development process, which shed light on the challenges along the way and how to overcome them. We start from early observations and clues, which lead us to explore extending a construction of a CA\((2q^3 - 1; 3, q^2 + q + 1, q)\) given by Raaphorst, Moura, and Stevens (2014) to a construction of a strength-\(4\) covering array.

A systematic construction we designed seemed to always lead to a CA\((3q^4-2; 4, \frac{q^2 +1}{2}, q)\), when \(q\) was a prime power. While trying to develop a proof for the theorem, we examine possible approaches and finally focus on the geometric approach. The existence of three truncated M{"o}bius planes with restrictions on the intersection of their circles would pave our way to the final proof.

This is joint work with Lucia Moura and Brett Stevens.

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