MUBs from bent functions
This note contains a simple construction of complete sets of MUBs, using bent functions to write the new basis vectors as explicit linear combinations of the standard basis.
math.CO↗
arXiv subjects
William M. Kantor
Publications and source records attributed to William M. Kantor.
Soft planes and groups
Finite projective planes are constructed using groups that satisfy simple-looking conditions. The resulting projective planes include many known planes and possibly new ones, and are precisely those having a collineation group fixing a flag $(\infty,L_ \infty )$ and transitive on the flags $ (w,W )$ with $w\notin L_ \infty $ and $\infty\notin W$.
math.CO↗
Automorphism groups of Steiner triple systems
If $G$ is a finite group then there is an integer $M_G$ such that$,$ for $u\ge M_G$ and $u\equiv 1$ or $3$ (mod 6), there is a Steiner triple system $U$ on $u$ points for which ${\rm Aut} U \cong G. \ $ If $V$ is a Steiner triple system then there is an integer $N_V$ such that$,$ for $u\ge N_V$ and $u\equiv 1$ or $3$ $($mod $6),$ there is a Steiner triple system $U$ on $u$ points having $V$ as an ${\rm Aut} U$-invariant subsystem such that ${\rm Aut} U\cong{\rm Aut} V $ and ${\rm Aut} U $ induces ${\rm Aut} V$ on $V$.
math.CO↗
On 2-transitive sets of equiangular lines
All finite sets of equiangular lines spanning finite-dimensional unitary spaces are determined for which the action on the lines of the set-stabilizer in the unitary group is 2-transitive with a regular normal subgroup.
math.MG↗
Automorphism subgroups for designs with $λ=1$
Given an integer $k\ge3$ and a group $G$ of odd order, if there exists a $2$-$(v,k,1)$-design and if $v$ is sufficiently large, then there is such a design whose automorphism group has a subgroup isomorphic to $G$. A weaker result is proved when $|G|$ is even and $(k,|G|)=1$.
math.CO↗
Automorphism groups of designs with $λ=1$
If $G$ is a finite group and $k =q>2$ or $k=q+1$ for a prime power $q$ then, for infinitely many integers $v$, there is a $2$-$(v,k,1)$-design ${\bf D}$ for which ${\rm Aut} {\bf D}\cong G$.
math.CO↗
Antiflag Transitive Collineation Groups
We present known results concerning antiflag transitive collineation groups of finite projective spaces and finite polar spaces.
math.CO↗
Invariable Generation of Infinite Groups
A subset S of a group G invariably generates G if G = for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a finitely generated linear group is invariably generated by some finite set of elements if and only if it is virtually solvable. We also show that the profinite completion of an arithmetic group having the congruence subgroup property is invariably generated by a finite set of elements.
math.GR↗
Orthogonal Dual Hyperovals, Symplectic Spreads and Orthogonal Spreads
Orthogonal spreads in orthogonal spaces of type $V^+(2n+2,2)$ produce large numbers of rank $n$ dual hyperovals in orthogonal spaces of type $V^+(2n,2)$. The construction resembles the method for obtaining symplectic spreads in $V(2n,q)$ from orthogonal spreads in $V^+(2n+2,q)$ when $q$ is even.
math.CO↗
Bent functions generalizing Dillon's partial spread functions
This note presents generalizations of the partial spread bent functions introduced by Dillon, as well as the corresponding relative difference sets in nonabelian groups.
math.CO↗
A projection construction for semifields
Simple constructions are given for finite semifields that include as special cases both old semifields and recently constructed semifields.
math.CO↗
Presentations of Finite Simple Groups: Profinite and Cohomological Approaches
We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt.
math.GR↗