Triple arrays and Youden squares

Abstract

This paper addresses the question of when triple arrays can be constructed from Youden squares by removing a column together with the symbols therein, and then exchanging the role of columns and symbols. The scope of the investigation is limited to the standard case of triple arrays with \(v=r+c-1\). For triple arrays with \(\lambda _{cc}=1\) it is proven that they can never be constructed in this way, and for triple arrays with \(\lambda _{cc}=2\) it is proven that there always exists a suitable Youden square and a suitable column for this construction. Further, it is proven that Youden square constructed from a certain family of difference sets never give rise to triple arrays in this way but always gives rise to double arrays. Finally, it is proven that all triple arrays in the single known infinite family, the Paley triple arrays, can all be constructed in this way for some suitable choice of Youden square and column.