Extending the $t$-design concept

Abstract:

Let $\mathfrak {B}$ be a family of $k$-subsets of a $v$-set $V$, with $1 \leq k \leq v/2$. Given only the inner distribution of $\mathfrak {B}$, i.e., the number of pairs of blocks that meet in $j$ points (with $j = 0,1, \ldots ,k$), we are able to completely describe the regularity with which $\mathfrak {B}$ meets an arbitrary $t$-subset of $V$, for each order $t$ (with $1 \leq t \leq v/2$). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters $v$, $k$, $t$. The main regularity parameter is the dimension of a well-defined subspace of ${\mathbb {R}^{t + 1}}$, called the $t$-form space of $\mathfrak {B}$. (This subspace coincides with ${\mathbb {R}^{t + 1}}$ if and only if $\mathfrak {B}$ is a $t$-design.) We show that the $t$-form space has the structure of an ideal, and we explain how to compute its canonical generator.