114 research outputs found
On the number of n-ary quasigroups of finite order
Let be the number of -ary quasigroups of order . We derive a
recurrent formula for Q(n,4). We prove that for all and the
following inequalities hold: , where does not depend on . So, the upper
asymptotic bound for is improved for any and the lower bound
is improved for odd . Keywords: n-ary quasigroup, latin cube, loop,
asymptotic estimate, component, latin trade.Comment: english 9pp, russian 9pp. v.2: corrected: initial data for recursion;
added: Appendix with progra
On the number of frequency hypercubes
A frequency -cube is an -dimensional array filled by s and s such that each line contains exactly two s.
We classify the frequency -cubes , find a testing set of size
for , and derive an upper bound on the number of .
Additionally, for any greater than , we construct an that
cannot be refined to a latin hypercube, while each of its sub-
can.
Keywords: frequency hypercube, frequency square, latin hypercube, testing
set, MDS cod
Maximal partial Latin cubes
We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders n the numbers k for which there exists a maximal partial Latin cube of order n with exactly k entries. Finally, we prove that maximal partial Latin cubes of order n exist of each size from approximately half-full (n3/2 for even n ≥ 10 and (n3 + n)/2 for odd n ≥21) to completely full, except for when either precisely 1 or 2 cells are empty
Multi-latin squares
A multi-latin square of order and index is an array of
multisets, each of cardinality , such that each symbol from a fixed set of
size occurs times in each row and times in each column. A
multi-latin square of index is also referred to as a -latin square. A
-latin square is equivalent to a latin square, so a multi-latin square can
be thought of as a generalization of a latin square.
In this note we show that any partially filled-in -latin square of order
embeds in a -latin square of order , for each , thus
generalizing Evans' Theorem. Exploiting this result, we show that there exist
non-separable -latin squares of order for each . We also show
that for each , there exists some finite value such that for
all , every -latin square of order is separable.
We discuss the connection between -latin squares and related combinatorial
objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares
and -latin trades. We also enumerate and classify -latin squares of small
orders.Comment: Final version as sent to journa
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