A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q odd

In this article, we prove a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q odd, i.e. for every integer k in the interval [a, b], where a approximate to q2 and b approximate to 9/10q2, there exists a maximal partial ovoid of Q(4, q), q odd, of size k. Since the generalized quadrangle IN(q) defined by a symplectic polarity of PG(3, q) is isomorphic to the dual of the generalized quadrangle Q(4, q), the same result is obtained for maximal partial spreads of 1N(q), q odd. This article concludes a series of articles on spectrum results on maximal partial ovoids of Q(4, q), on spectrum results on maximal partial spreads of VV(q), on spectrum results on maximal partial 1-systems of Q(+)(5,q), and on spectrum results on minimal blocking sets with respect to the planes of PG(3, q). We conclude this article with the tables summarizing the results

Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of unextendible mutually unbiased bases. We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set of nonidentity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d a prime, and the geometry of non-degenerate alternating bilinear forms of rank N over finite fields F d We then supply alternative and short proofs of results obtained in Mandayam et al. (2014), as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of Mandayam et al. (2014) and speculate on variations of this conjecture

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]

On the smallest maximal partial ovoids and spreads of the generalized quadrangles W(q) and Q(4, q)

AbstractWe present results on the size of the smallest maximal partial ovoids and on the size of the smallest maximal partial spreads of the generalized quadrangles W(q) and Q(4,q)