Multi-letter Youden rectangles from quadratic forms (Q1810639)

A multi-letter Youden rectangle is a set \(C\) of \(vk\) cells, with a partition \({\mathcal A}\) of \(C\) into \(k\) sets of size \(v\), and \(r\) partitions \({\mathcal B}_1, \dots,{\mathcal B}_r\) of \(C\) into \(v\) sets of size \(k\), satisfying: each part of \({\mathcal A}\) meets each part of \({\mathcal B}_i\) in one cell; each part of \({\mathcal B}_i\) meets each part of \({\mathcal B}_j\) in at most one cell (where non-empty intersection implies two sets are incident); and the sets \({\mathcal B}_1, \dots,{\mathcal B}_r\) with the above incidence relation, form a system of linked symmetric designs. The author uses quadratic forms to construct these proving that there exists a multi-letter Youden rectangle with \(v=2^{2n}\), \(k=2^{2n-1} \pm 2^{n-1},\) and \(r=2^n\), for all \(n \geq 2.\)