Related papers: Singer 8-arcs of Mathon type in PG(2,2^7)
A $2$-semiarc is a pointset ${\mathcal S}_k$ with the property that the number of tangent lines to ${\mathcal S}_k$ at each of its points is two. Using some theoretical results and computer aided search, the complete classification of…
We characterise finite groups such that for an odd prime $p$ all the irreducible characters in its principal $p$-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by $p$ unless…
In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence…
A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a…
We construct two-weight sets in PG$(3n-1,q)$, $n\geq2$ with the same weights as those that would arise from the blow-up of a maximal $q$-arc in PG$(2,q^n)$. The construction is of particular interest when $q$ is odd, as it is well known…
It is known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7/2*k^2-9/4*k+3/2$ even ovals. We show here that this upper bound is asymptotically sharp, that is to say we…
The question of existence of a maximal subgroup in the multiplicative group D* of a division algebra D finite dimensional over its center F is investigated. We prove that if D* has no maximal subgroup, then deg(D) is not a power of 2,…
A $k$-arc in PG($2,q$) is a set of $k$ points no three of which are collinear. A hyperfocused $k$-arc is a $k$-arc in which the $k \choose 2$ secants meet some external line in exactly $k-1$ points. Hyperfocused $k$-arcs can be viewed as…
We analyze the group structures of two groups of order 1344 which are respectively non-split and split extensions of the elementary Abelian group of order 8 by its automorphism group PSL_2(7).They share the same character table. The group…
In this paper, we propose an algorithm to enumerate genus-4 superspecial hyperelliptic curves whose automorphism groups isomorphic to the quaternion group. By implementing this algorithm with Magma, we successfully obtain the number of…
In this paper we continue our program, started in [2], of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for E7, by first providing explicit matrix…
We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and…
We demonstrate a close connection between the classic planar Singer difference sets and certain norm equation systems arising from projective norm graphs. This, on the one hand leads to a novel description of planar Singer difference sets…
Two variants of $N=7$ superconformal mechanics with manifest $so(7)$ and $su(2) \times su(2)$ $R$-symmetries possessing exceptional $G(3)$ dynamical symmetry are presented. To construct these $G(3)$ theories, we developed two new versions…
A finite generalized quadrangle $\cS$ is a Singer quadrangle if it has an automorphism group that acts sharply transitively on its points. In this paper, we introduce the notion of multipliers for a Singer quadrangle and study their basic…
We give a construction that takes a simple linear algebraic group $G$ over a field and produces a commutative, unital, and simple non-associative algebra $A$ over that field. Two attractions of this construction are that (1) when $G$ has…
We show that the largest character degree of an alternating group $A_n$ with $n\geq 5$ can be bounded in terms of smaller degrees in the sense that \[ b(A_n)^2<\sum_{\psi\in\textrm{Irr}(A_n),\,\psi(1)< b(A_n)}\psi(1)^2, \] where…
Up to isomorphism and duality, there are exactly two non-degenerate abstract regular polytopes of rank greater than $n-3$, one of rank $n-1$ and one of rank $n-2$, with automorphism groups that are transitive permutation groups of degree…
In this paper, we describe minimal presentations of maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type. For this purpose, we introduce a new class of pro-$2$ groups: $\Delta$-Right Angled…
In \cite{M18}, the first author gave a construction of strongly regular Cayley graphs on the additive group of finite fields by using three-valued Gauss periods. In particular, together with the result in \cite{BLMX}, it was shown that…