Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (qⁿ,qⁿ) and a Laguerre plane of order qⁿ (for some n). In setting out a programme to construct new generalised quadrangles, Shult and Thas [15] asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q⁻(5,q), non-equivalent to the classical example, a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q⁻(5,q) is projectively equivalent to a pseudo-conic. Thas [16] characterised pseudo-conics as pseudo-ovals satisfying the perspective property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q⁻(5,q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. These association schemes are introduced and explored, and we provide a complete theory of how pseudo-ovals of lines of Q⁻(5,q) can be analysed from this viewpoint.