Constructing prime-field planar configurations

An infinite class of planar configurations is constructed with distinct prime-field characteristic sets (i.e., configurations represented over a finite set of prime fields but over fields of no other characteristic). It is shown that if $p$ is sufficiently large, then every subset of $k$ primes between $p$ and $f(p,k)$ forms such a set (where $f(p,k) = {2^{[(\sqrt p - A{k^{3/2}})/B{k^{3/2}}]}}$ for constants $A$ and $B$). In particular, for every positive integer $k$, there exist infinitely many planar matroid configurations ${C_{i,k}}$ with $\left\vert {{\chi _{pf}}({C_{i,k}})} \right\vert = k$ (where ${\chi _{pf}}(C)$ denotes the prime-field characteristic set of $C$). We also give a result concerning cofinite prime-field characteristic sets.