1 research outputs found
On generalized Howell designs with block size three
In this paper, we examine a class of doubly resolvable combinatorial objects.
Let and be nonnegative integers, and let be a set of
symbols. A generalized Howell design, denoted -,
is an array, each cell of which is either empty or contains a
-set of symbols from , called a block, such that: (i) each symbol appears
exactly once in each row and in each column (i.e.\ each row and column is a
resolution of ); (ii) no -subset of elements from appears in more
than cells. Particular instances of the parameters correspond to
Howell designs, doubly resolvable balanced incomplete block designs (including
Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple
orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin
squares). Generalized Howell designs also have connections with permutation
arrays and multiply constant-weight codes.
In this paper, we concentrate on the case that , and ,
and write . In this case, the number of empty cells in each row and
column falls between 0 and . Previous work has considered the
existence of GHDs on either end of the spectrum, with at most 1 or at least
empty cells in each row or column. In the case of one empty cell, we
correct some results of Wang and Du, and show that there exists a
if and only if , except possibly for . In the case of two empty
cells, we show that there exists a if and only if .
Noting that the proportion of cells in a given row or column of a
which are empty falls in the interval , we prove that for any , there is a whose proportion of empty cells in a row or
column is arbitrarily close to .Comment: 31 pages; 1 figure; 2 appendice