Sets of mutually orthogonal Latin squares with ``like subsquares''
The concepts of ``transversal identifying Latin squares'' and Latin squares with ``like subsquares'' are introduced. The transversal identifying property permits the determination of orthogonality in computing time which is proportional to the cube of the order of Latin squares instead of the fourth power. With the aid of his method, using a computer, the author has obtained orthogonal sets of Latin squares of orders \(n=12, 15, 20\), and \(21\). The numbers of Latin squares of the orthogonal sets obtained by the author are equal to the current lower bounds. In the case when \(n=24\) the known value four has been improved to five. Reviewer's remarks: The efficiency of the author's algorithm, when two Latin squares are given and one would like to know are they orthogonal or not is far from being optimal (as far as the reviewer can see). The efficiency of the author's algorithm when it is used to construct sets of mutually orthogonal Latin squares is very poor. One can find more efficient algorithms on page 168 of the recent book ``Latin squares. New developments in the theory and applications'' edited by the reviewer and \textit{A. D. Keedwell} [Ann. Discrete Math. 46 (1991; Zbl 0715.00010)].
scientific article; zbMATH DE number 68484
scientific article