  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 $(q^n,q^n)$ and a
Laguerre plane of order $q^n$ (for some $n$). In setting out a programme to
construct new generalised quadrangles, Shult and Thas 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 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.
