# Codes and Expansions (CodEx) Seminar

## Jonathan Jedwab (Simon Fraser University)Perfect sequence covering arrays

An $(n,k,\lambda)$ perfect sequence covering array is a multiset whose elements are permutations of the sequence $(1, 2, \dots, n)$ and which collectively contain each ordered $k$-subsequence exactly $\lambda$ times. The central question is: for given $n$ and $k$, what is the smallest value of $\lambda$ (denoted $g(n,k)$) for which such a configuration exists? We interpret the sequences of a perfect sequence covering array as elements of the symmetric group $S_n$, and constrain its structure to be a union of cosets of a prescribed subgroup of $S_n$. By adapting a search algorithm due to Mathon and van Trung for finding spreads, we obtain highly structured examples of perfect sequence covering arrays. In particular, we determine that $g(6,3) = g(7,3) = g(7,4) = 2$ and $g(7,5) \in \{2,3,4\}$ and $g(8,3) \in \{2,3\}$ and $g(9,3) \in \{2,3,4\}$.

This is joint work with Jingzhou Na and Shuxing Li.