Multi-letter Youden rectangles from quadratic forms

Some infinite families of systems of linked symmetric designs (or SLSDs, for short) were constructed by Cameron and Seidel (Proc. Kon. Nederl. Akad. Wetensch. (A) 76 (1973) 1-8) using quadratic and bilinear forms over GF(2). The smallest of these systems was used by Preece and Cameron (Utilitas Math. 8 (1975) 193-204) to construct certain designs (which they called fully balanced hyper-graeco-latin Youden 'squares'). The purpose of this paper is to construct an infinite sequence of closely related designs (here called multi-letter Youden rectangles) from the SLSDs of Cameron and Seidel. These rectangles are k×v, with v=2 ²ⁿ and k=2 ^2n-1±2 ^n-1. The paper also provides a non-trivial example of how to translate from the combinatorial view of designs (sets with incidence relations) to the statistical (sets with partitions).