4 research outputs found
More nonexistence results for symmetric pair coverings
A -covering is a pair , where is a
-set of points and is a collection of -subsets of
(called blocks), such that every unordered pair of points in is contained
in at least blocks in . The excess of such a covering is
the multigraph on vertex set in which the edge between vertices and
has multiplicity , where is the number of blocks which
contain the pair . A covering is symmetric if it has the same number
of blocks as points. Bryant et al.(2011) adapted the determinant related
arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the
nonexistence of certain symmetric coverings with -regular excesses. Here, we
adapt the arguments related to rational congruence of matrices and show that
they imply the nonexistence of some cyclic symmetric coverings and of various
symmetric coverings with specified excesses.Comment: Submitted on May 22, 2015 to the Journal of Linear Algebra and its
Application
Parity of Sets of Mutually Orthogonal Latin Squares
Every Latin square has three attributes that can be even or odd, but any two
of these attributes determines the third. Hence the parity of a Latin square
has an information content of 2 bits. We extend the definition of parity from
Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the
corresponding orthogonal arrays (OA). Suppose the parity of an
has an information content of bits. We show that
. For the case corresponding to projective
planes we prove a tighter bound, namely when
is odd and when is even. Using the
existence of MOLS with subMOLS, we prove that if
then for all sufficiently large .
Let the ensemble of an be the set of Latin squares derived by
interpreting any three columns of the OA as a Latin square. We demonstrate many
restrictions on the number of Latin squares of each parity that the ensemble of
an can contain. These restrictions depend on and
give some insight as to why it is harder to build projective planes of order than for . For example, we prove that when it is impossible to build an for which all
Latin squares in the ensemble are isotopic (equivalent to each other up to
permutation of the rows, columns and symbols)