Enumeration of Latin Squares and Isomorphism Detection in Finite Planes

摘要

In Chapter I, various necessary conditions for isomorphism between latin rectangles are developed by treating the rectangles as sets of permutations, and studying the cycle structure of the permutations. An enumeration procedure for latin squares, which operates by forming successively higher order rectangle representatives, is then given, and the theory is applied to latin squares of order 8, yielding 1,676,257 representative squares. Chapter II details a reversible process for obtaining a complete set of orthogonal squares from a finite projective plane of the same order, and an algorithm for determining when two projective planes, in orthogonal squares form, are isomorphic is then given. The algorithm operates by constructing a minimal set of orthogonal sets for each known plane, and testing some one orthogonal set from a prospective new plane against them. Chapter III discusses known methods of generating projective planes of order n from latin squares of order n - 1. (Author)