2021/04/16 by John Bamberg, Bamberg, John, Giusy Monzillo +3 · 1 citation
Computer Science · Engineering · Mathematics · #05E30 #51A50 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · doi:10.48550/arxiv.2104.07999
openalex publication_date 2021/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.