TY - JOUR

T1 - Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

AU - Bamberg, John

AU - Monzillo, Giusy

AU - Siciliano, Alessandro

PY - 2021/9/1

Y1 - 2021/9/1

N2 - 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.

AB - 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.

KW - Association scheme

KW - Elliptic quadric

KW - Pseudo-oval

UR - http://www.scopus.com/inward/record.url?scp=85105694019&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2021.04.014

DO - 10.1016/j.laa.2021.04.014

M3 - Article

AN - SCOPUS:85105694019

SN - 0024-3795

VL - 624

SP - 281

EP - 317

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -