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A $pseudo$-$oval$ of a finite projective space over a finite field of oddorder $q$ is a configuration of equidimensional subspaces that is essentiallyequivalent to a translation generalised quadrangle of order $(q^n,q^n)$ and aLaguerre plane of order $q^n$ (for some $n$). In setting out a programme toconstruct new generalised quadrangles, Shult and Thas asked whether there arepseudo-ovals consisting only of lines of an elliptic quadric ${Q}^-(5,q)$,non-equivalent to the $classical$ $example$, a so-called $pseudo$-$conic$. Todate, every known pseudo-oval of lines of ${Q}^-(5,q)$ is projectivelyequivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovalssatisfying the $perspective$ property, and this paper is on characterisationsof 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 characterisedas certain Delsarte designs of an interesting five-class association scheme.These association schemes are introduced and explored, and we provide acomplete theory of how pseudo-ovals of lines of $Q^-(5,q)$ can be analysed fromthis viewpoint.

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