Some new infinite series of Freeman-Youden rectangles (Q2761043)

A scheme of \(r\) rows that permute \(t\) symbols so that columns yield an SBIBD (i.e. a symmetric 2-design) is known as an \(r\times t\) Youden square. Consider two superimposed Youden squares using different symbol sets \(S1\) and \(S2\). Each of them yields a \(t\times t\) incidence matrix of an SBIBD; denote the matrices by \(n_{01}\) and \(n_{02}\). Relate \(s1 \in S1\) and \(s2 \in S2\), if they are superimposed at some position. If this relation yields an incidence matrix \(n_{12}\) of another SBIBD, and if \(n_{01}n_{12}n_{20}+n_{02}n_{21}n_{10}=fI+gJ \) holds for some integers \(f\) and \(g\), then the superimposition is called balanced (reversal of indices indicates transposition, \(I\) is the identity matrix and \(J\) the all-one matrix). The authors suggest to call such a structure a Freeman-Youden rectangle (FYR). In the paper they construct an infinite series of \(q\times (2q+1)\) FYRs, \(q>3\) a prime power with \(q\equiv 3 \bmod 4\), and describe certain variations of the construction. The construction uses a \(q\times q\) matrix with entries \(0\) and \(1\), where the \(ij\)th entry equals 1 if \(i-j\) is an even power of a fixed primitive element of \(\text{GF}(q)\).