Perfect binary arrays and difference sets
Ingenuous elementary constructions are given to get perfect binary arrays \[ \text{PBA}(2^ y,2^ y),\quad \text{PBA}(2^{y+1}, 2^{y-1}), \quad \text{PBA}(2^ y\cdot 3, 2^ y\cdot 3), \quad \text{and} \quad \text{PBA}(2^{y+1}\cdot 3, 2^{y-1}\cdot 3). \] The basic idea is to use particular compositions of a \(\text{PBA}(s,t)\) and a ``rowwise quasiperfect'' array \(s\times t\) and iterate this or a similar construction. As bonus, the \((4n^ 2, 2n^ 2- n, n^ 2- n)\)-difference sets equivalent to \(\text{PBA}(2^ y, 2^ y)\) \((2^{2y}= 4n^ 2)\) is fixed by the multiplier \(-1\), and quasiperfect arrays with interesting symmetries are obtained. See, e.g., \textit{D. Calabro} and \textit{J. K. Wolf} [On the synthesis of two-dimensional arrays with desirable correlation properties, Inf. Control 11 (1967), 537-560 (1968; Zbl 0157.494)] for sources and motivations (also linked to Singer groups) of the questions involved: unfortunately many of the quoted papers are not so easy to find in a mathematical department.
scientific article; zbMATH DE number 562668
scientific article