The existence of howell designs of siden+1 and order 2n

Abstract

AHowell design of side s andorder 2n, or more briefly, anH(s, 2n), is ans×s array in which each cell either is empty or contains an unordered pair of elements from some 2n-set, sayX, such that

(a) each row and each column is Latin (that is, every element ofX is in precisely one cell of each row and each column) and

(b) every unordered pair of elements fromX is in at most one cell of the array. Atrivial Howell design is anH(s, 0) havingX=Ø and consisting of ans×s array of empty cells. A necessary condition onn ands for the existence of a nontrivialH(s, 2n) is that 0<n≦s≦2n-1.

AnH(n+t, 2n) is said to contain a maximum trivial subdesign if somet×t subarray is theH(t, 0). This paper describes a recursive construction for Howell designs containing maximum trivial subdesigns and applies it to settle the existence question forH(n+1, 2n)’s: forn+1 a positive integer, there is anH(n+1, 2n) if and only ifn+1 ∉ {2, 3, 5}.