4,609 research outputs found
Maximal integral point sets in affine planes over finite fields
Motivated by integral point sets in the Euclidean plane, we consider integral
point sets in affine planes over finite fields. An integral point set is a set
of points in the affine plane over a finite field
, where the formally defined squared Euclidean distance of every
pair of points is a square in . It turns out that integral point
sets over can also be characterized as affine point sets
determining certain prescribed directions, which gives a relation to the work
of Blokhuis. Furthermore, in one important sub-case integral point sets can be
restated as cliques in Paley graphs of square order. In this article we give
new results on the automorphisms of integral point sets and classify maximal
integral point sets over for . Furthermore, we give two
series of maximal integral point sets and prove their maximality.Comment: 18 pages, 3 figures, 2 table
Integral point sets over finite fields
We consider point sets in the affine plane where each
Euclidean distance of two points is an element of . These sets
are called integral point sets and were originally defined in -dimensional
Euclidean spaces . We determine their maximal cardinality
. For arbitrary commutative rings
instead of or for further restrictions as no three points on a
line or no four points on a circle we give partial results. Additionally we
study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure
Markoff Triples and Strong Approximation
We investigate the transitivity properties of the group of morphisms
generated by Vieta involutions on the solutions in congruences to the Markoff
equation as well as to other Markoff type affine cubic surfaces. These are
dictated by the finite orbits of these actions and these can
be determined effectively. The results are applied to give forms of strong
approximation for integer points, and to sieving, on these surface
Symplectic spreads, planar functions and mutually unbiased bases
In this paper we give explicit descriptions of complete sets of mutually
unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras
obtained from commutative and symplectic semifields, and
from some other non-semifield symplectic spreads. Relations between various
constructions are also studied. We show that the automorphism group of a
complete set of MUBs is isomorphic to the automorphism group of the
corresponding orthogonal decomposition of the Lie algebra .
In the case of symplectic spreads this automorphism group is determined by the
automorphism group of the spread. By using the new notion of pseudo-planar
functions over fields of characteristic two we give new explicit constructions
of complete sets of MUBs.Comment: 20 page
Tropical Skeletons
In this paper, we study the interplay between tropical and analytic geometry
for closed subschemes of toric varieties. Let be a complete non-Archimedean
field, and let be a closed subscheme of a toric variety over . We define
the tropical skeleton of as the subset of the associated Berkovich space
which collects all Shilov boundary points in the fibers of the
Kajiwara--Payne tropicalization map. We develop polyhedral criteria for limit
points to belong to the tropical skeleton, and for the tropical skeleton to be
closed. We apply the limit point criteria to the question of continuity of the
canonical section of the tropicalization map on the multiplicity-one locus.
This map is known to be continuous on all torus orbits; we prove criteria for
continuity when crossing torus orbits. When is sch\"on and defined over a
discretely valued field, we show that the tropical skeleton coincides with a
skeleton of a strictly semistable pair, and is naturally isomorphic to the
parameterizing complex of Helm--Katz.Comment: 42 pages. The introduction was rewritten. Corollary 8.15 was renamed
to Theorem 8.1
Maximum Distance Separable Codes and Arcs in Projective Spaces
Given any linear code over a finite field we show how can be
described in a transparent and geometrical way by using the associated
Bruen-Silverman code. Then, specializing to the case of MDS codes we use our
new approach to offer improvements to the main results currently available
concerning MDS extensions of linear MDS codes. We also sharply limit the
possibilities for constructing long non-linear MDS codes.Comment: 18 Pages; co-author added; some results updated; references adde
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