Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in dimension at most $4$. Techniques are given for determining when two solutions are linearly conformally inequivalent.Comment: v3. Typos correcte

The classification of flag-transitive Steiner 3-designs

We solve the long-standing open problem of classifying all 3-(v,k,1) designs with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom. Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F. Buekenhout, Elsevier Science, Amsterdam, 1995, p. 273; but presumably dating back to 1965). Our result relies on the classification of the finite 2-transitive permutation groups.Comment: 27 pages; to appear in the journal "Advances in Geometry

Classification of Cyclic Steiner Quadruple Systems

The problem of classifying cyclic Steiner quadruple systems (CSQSs) is considered. A computational approach shows that the number of isomorphism classes of such designs with orders 26 and 28 is 52,170 and 1,028,387, respectively. It is further shown that CSQSs of order 2p, where p is a prime, are isomorphic iff they are multiplier equivalent. Moreover, no CSQSs of order less than or equal to 38 are isomorphic but not multiplier equivalent

Harmonic equiangular tight frames comprised of regular simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

A quasidouble of the affine plane of order 4 and the solution of a problem on additive designs

A 2-(v,k,λ) block design (P,B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G,+) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G, the embedding of P in G is also such that, conversely, any zero-sum k-subset of P is a block in B, then (P,B) is said to be strongly additive. In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2-(16,4,2) design (F_4×F_4, B_2), which decomposes into two disjoint isomorphic copies of the affine plane of order four. An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG(4,2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem