454 research outputs found
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
PARTITIONING THE BLOCKS OF A STEINER TRIPLE SYSTEM INTO PARTIAL PARALLEL CLASSES
Does there exist a Steiner Triple System on v points, whose blocks can be partitioned into partial parallel classes of size m, where m ≤ [v⁄3], m | b and b is the number of blocks of the STS(v)? We give the answer for 9 ≤ v ≤ 43. We also show that whenever 2|b, v ≡ 3 (mod 6) we can find an STS(v) whose blocks can be partitioned into partial parallel classes of size 2, and whenever 4|b , v ≡ 3 (mod 6), there exists an STS(v) whose blocks can be partitioned into partial parallel classes of size 4
Distributive and trimedial quasigroups of order 243
We enumerate three classes of non-medial quasigroups of order up to
isomorphism. There are non-medial trimedial quasigroups of order
(extending the work of Kepka, B\'en\'eteau and Lacaze), non-medial
distributive quasigroups of order (extending the work of Kepka and
N\v{e}mec), and non-medial distributive Mendelsohn quasigroups of order
(extending the work of Donovan, Griggs, McCourt, Opr\v{s}al and
Stanovsk\'y).
The enumeration technique is based on affine representations over commutative
Moufang loops, on properties of automorphism groups of commutative Moufang
loops, and on computer calculations with the \texttt{LOOPS} package in
\texttt{GAP}
Selected Papers in Combinatorics - a Volume Dedicated to R.G. Stanton
Professor Stanton has had a very illustrious career. His contributions to mathematics are varied and numerous. He has not only contributed to the mathematical literature as a prominent researcher but has fostered mathematics through his teaching and guidance of young people, his organizational skills and his publishing expertise. The following briefly addresses some of the areas where Ralph Stanton has made major contributions
- …