36 research outputs found
The fine intersection problem for Steiner triple systems
The intersection of two Steiner triple systems (X,A) and (X,B) is the set A
intersect B. The fine intersection problem for Steiner triple systems is to
determine for each v, the set I(v), consisting of all possible pairs (m,n) such
that there exist two Steiner triple systems of order v whose intersection has n
blocks over m points. We show that for v = 1 or 3 (mod 6), |I(v)| = Omega(v^3),
where previous results only imply that |I(v)| = Omega(v^2).Comment: 9 page
Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three
The concept of group divisible codes, a generalization of group divisible
designs with constant block size, is introduced in this paper. This new class
of codes is shown to be useful in recursive constructions for constant-weight
and constant-composition codes. Large classes of group divisible codes are
constructed which enabled the determination of the sizes of optimal
constant-composition codes of weight three (and specified distance), leaving
only four cases undetermined. Previously, the sizes of constant-composition
codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table
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