38 research outputs found
On arcs in projective Hjelmslev planes
AbstractA (k,n)-arc in the projective Hjelmslev plane PHG(RR3) is defined as a set of k points in the plane such that some n but no n+1 of them are collinear. In this paper, we consider the problem of finding the largest possible size of a (k,n)-arc in PHG(RR3). We present general upper bounds on the size of arcs in the projective Hjelmslev planes over chain rings R with |R|=q2,R/radR≅Fq. We summarize the known values and bounds on the cardinalities of (k,n)-arcs in the chain rings with |R|⩽25(|R|=q2,R/radR≅Fq)
An Algorithm for constructing Hjelmslev planes
Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations
of projective planes and affine planes. We present an algorithm for
constructing a projective Hjelmslev planes and affine Hjelsmelv planes using
projective planes, affine planes and orthogonal arrays. We show that all
2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv
planes can be constructed in this way. As a corollary it is shown that all
2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective
Hjelmselv planes.Comment: 15 pages. Algebraic Design Theory and Hadamard matrices, 2014,
Springer Proceedings in Mathematics & Statistics 13
Hjelmslev Geometry of Mutually Unbiased Bases
The basic combinatorial properties of a complete set of mutually unbiased
bases (MUBs) of a q-dimensional Hilbert space H\_q, q = p^r with p being a
prime and r a positive integer, are shown to be qualitatively mimicked by the
configuration of points lying on a proper conic in a projective Hjelmslev plane
defined over a Galois ring of characteristic p^2 and rank r. The q vectors of a
basis of H\_q correspond to the q points of a (so-called) neighbour class and
the q+1 MUBs answer to the total number of (pairwise disjoint) neighbour
classes on the conic.Comment: 4 pages, 1 figure; extended list of references, figure made more
illustrative and in colour; v3 - one more figure and section added, paper
made easier to follow, references update
CAPS in Z(2,n)
We consider point sets in (Z^2,n) where no three points are on a
line – also called caps or arcs. For the determination of caps with maximum
cardinality and complete caps with minimum cardinality we provide integer
linear programming formulations and identify some values for small n