On generalized Howell designs with block size three

TL;DR

This paper studies generalized Howell designs with block size three, establishing existence conditions for certain parameters and exploring the range of empty cell proportions in these combinatorial structures.

Contribution

It provides new existence results for GHDs with specific empty cell counts and characterizes the possible proportions of empty cells in these designs.

Findings

Existence of GHD(n+1,3n) for n ≥ 6, except possibly n=6.

Existence of GHD(n+2,3n) for n ≥ 6.

Range of empty cell proportions in GHDs can approach any value in [0,5/18].

Abstract

In this paper, we examine a class of doubly resolvable combinatorial objects. Let $t, k, \lambda, s$ and $v$ be nonnegative integers, and let $X$ be a set of $v$ symbols. A generalized Howell design, denoted $t$-$GHD_{k}(s,v;\lambda)$, is an $s\times s$ array, each cell of which is either empty or contains a $k$-set of symbols from $X$, 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 $X$); (ii) no $t$-subset of elements from $X$ appears in more than $\lambda$ 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 $t=2$, $k=3$ and $\lambda=1$, and write $GHD(s,v)$. In this case, the number of empty cells in each row and column falls between 0 and $(s-1)/3$. Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least $(s-2)/3$ 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 $GHD(n+1,3n)$ if and only if $n \geq 6$, except possibly for $n=6$. In the case of two empty cells, we show that there exists a $GHD(n+2,3n)$ if and only if $n \geq 6$. Noting that the proportion of cells in a given row or column of a $GHD(s,v)$ which are empty falls in the interval $[0,1/3)$, we prove that for any $\pi \in [0,5/18]$, there is a $GHD(s,v)$ whose proportion of empty cells in a row or column is arbitrarily close to $\pi$.