A counter-example to a conjecture relating complete sets of frequency squares and affine geometries
One of the most important results in combinatorics is the result from 1938 by R. C. Bose which proves the equivalence between complete sets of mutually orthogonal latin squares of order \(n\) and affine planes of order \(n\). The affine geometry of dimension \(d \geq 2\) is a natural generalization of an affine plane. In recent years numerous papers have been written concerning frequency squares which are generalizations of latin squares in which repetitions are allowed in each row and column, see for example, G. L. Mullen [Discrete Math. 69, No. 1, 79-84 (1988; Zbl 0638.05010)]. A natural generalization of the Bose result would be an equivalence between complete sets of mutually orthogonal frequency squares and higher dimensional affine geometries. Unfortunately as the author of the present paper shows, the most natural equivalence does always not hold. The problem of when such an equivalence does hold has been resolved in \textit{C. F. Laywine} and \textit{G. L. Mullen} [J. Comb. Theory, Ser. A 61, No. 1, 13-35 (1992; Zbl 0760.05011)]. See this paper p. 26 for an explanation of why the example of the paper under review has the stated properties.
scientific article; zbMATH DE number 500642