Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares

In this paper we consider the problem of enumerating diagonal Latin squares of small order. In particular we discuss possible algorithmic approaches to this problem and show our results in this regard. Surprisingly, our research showed that the best algorithm for enumerating diagonal Latin squares consists of a number of fixed loops, and its effectiveness can be significantly increased by careful tuning and applying special heuristics. We used the constructed algorithm to enumerate all diagonal Latin squares of order 8. Also, it is being used to carry out large-scale computational experiment aimed at enumeration of diagonal Latin squares of order 9.