15 research outputs found
Cyclotomic orthomorphisms of finite fields
AbstractIn this paper, we give explicit formulas for the number of cyclotomic orthomorphisms of Fq of index 3,4,5,6 for certain classes of prime powers q. We also give an explicit formula for the number of cyclotomic mappings of index 2 that are strong complete mappings of Fp for prime p
Row-Hamiltonian Latin squares and Falconer varieties
A \emph{Latin square} is a matrix of symbols such that each symbol occurs
exactly once in each row and column. A Latin square is
\emph{row-Hamiltonian} if the permutation induced by each pair of distinct rows
of is a full cycle permutation. Row-Hamiltonian Latin squares are
equivalent to perfect -factorisations of complete bipartite graphs. For the
first time, we exhibit a family of Latin squares that are row-Hamiltonian and
also achieve precisely one of the related properties of being
column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct
non-trivial, anti-associative, isotopically -closed loop varieties, solving
an open problem posed by Falconer in 1970
Isomorphisms of quadratic quasigroups
Let be a finite field of odd order and
be such that and
, where is the extended quadratic character. Let
be the quasigroup upon defined by if , and if . We show that if and only if for some . We also
characterise and exhibit further properties, including
establishing when is a Steiner quasigroup or is commutative,
entropic, left or right distributive, flexible or semisymmetric. In proving our
results we also characterise the minimal subquasigroups of
Latin bitrades derived from quasigroup autoparatopisms
In 2008, Cavenagh and Dr\'{a}pal, et al, described a method of constructing
Latin trades using groups. The Latin trades that arise from this construction
are entry-transitive (that is, there always exists an autoparatopism of the
Latin trade mapping any ordered triple to any other ordered triple). Moreover,
useful properties of the Latin trade can be established using properties of the
group. However, the construction does not give a direct embedding of the Latin
trade into any particular Latin square. In this paper, we generalize the above
to construct Latin trades embedded in a Latin square , via the
autoparatopism group of the quasigroup with Cayley table . We apply this
theory to identify non-trivial entry-transitive trades in some group operation
tables as well as in Latin squares that arise from quadratic orthomorphism
Uncoverings on graphs and network reliability
We propose a network protocol similar to the -tree protocol of Itai and
Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we
define an {\em -uncovering-by-bases} for a connected graph to be a
collection of spanning trees for such that any -subset of
edges of is disjoint from at least one tree in , where is
some integer strictly less than the edge connectivity of . We construct
examples of these for some infinite families of graphs. Many of these infinite
families utilise factorisations or decompositions of graphs. In every case the
size of the uncovering-by-bases is no larger than the number of edges in the
graph and we conjecture that this may be true in general.Comment: 12 pages, 5 figure