On Howell designs

A Howell Design of type H(s, 2n) consists of a square of side s such that each cell is either empty or contains an unordered pair of integers taken from amongst 1, 2, 3, …, 2n provided: (1) each integer from 1 to 2n appears exactly once in each row and each column of the square and (2) every unordered pair appears at most once in a cell of the square. It is easily seen that for a Howell Design to exist that n ? s ? 2n ? 1. This paper presents a large number of constructions of Howell Designs and some existence theorems.