53 research outputs found
Two conjectures of Demetrovics, Füredi, and Katona, concerning partitions
AbstractIt is possible to find n partitions of an n-element set whose pairwise intersections are just all atoms of the partition lattice? Demetrovics, Füredi and Katona [4] verified this for all n ≡ 1 or 4 (mod 12) by constructing a series of special Mendelsohn Triple Systems. They conjectured that such triple systems exist for all n ≡ 1 (mod 3) and that the problem on the partitions has a solution for all n ⩾ 7. We prove that both conjectures are ture, except for finitely many n
The PBD-Closure of Constant-Composition Codes
We show an interesting PBD-closure result for the set of lengths of
constant-composition codes whose distance and size meet certain conditions. A
consequence of this PBD-closure result is that the size of optimal
constant-composition codes can be determined for infinite families of parameter
sets from just a single example of an optimal code. As an application, the size
of several infinite families of optimal constant-composition codes are derived.
In particular, the problem of determining the size of optimal
constant-composition codes having distance four and weight three is solved for
all lengths sufficiently large. This problem was previously unresolved for odd
lengths, except for lengths seven and eleven.Comment: 8 page
Mutually orthogonal latin squares with large holes
Two latin squares are orthogonal if, when they are superimposed, every
ordered pair of symbols appears exactly once. This definition extends naturally
to `incomplete' latin squares each having a hole on the same rows, columns, and
symbols. If an incomplete latin square of order has a hole of order ,
then it is an easy observation that . More generally, if a set of
incomplete mutually orthogonal latin squares of order have a common hole of
order , then . In this article, we prove such sets of
incomplete squares exist for all satisfying
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