Field reduction and linear sets in finite geometry

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields

'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that two of these sets/configurations, namely the $18_{2} - 12_{3}$ and $2_{4}14_{2} - 4_{3}6_{4}$ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types ${\cal V}_{22}(37; 0, 12, 15, 10)$ and ${\cal V}_{4}(49; 0, 0, 21, 28)$ in the classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797]. Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained

Qudits of composite dimension, mutually unbiased bases and projective ring geometry

The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may be mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$ ($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings $\mathcal{Z}_{6}$, $\mathcal{Z}_{10}$, $\mathcal{Z}_{15}$, $\mathcal{Z}_6 \times \mathbf{F}_4$ and $\mathcal{Z}_6 \times \mathcal{Z}_3$, respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A Mathematical and Theoretical (2008) accepte

An entropy for groups of intermediate growth

One of the few accepted dynamical foundations of non-additive "non-extensive") statistical mechanics is that the choice of the appropriate entropy functional describing a system with many degrees of freedom should reflect the rate of growth of its configuration or phase space volume. We present an example of a group, as a metric space, that may be used as the phase space of a system whose ergodic behavior is statistically described by the recently proposed $\delta$-entropy. This entropy is a one-parameter variation of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from the power-law entropies that have been recently studied. We use the first Grigorchuk group for our purposes. We comment on the connections of the above construction with the conjectured evolution of the underlying system in phase space.Comment: 19 pages, No figures, LaTeX2e. Version 2: change of affiliation, addition of acknowledgement. Accepted for publication to "Advances in Mathematical Physics