The Non-existence of (3,l,2)-Conjugate Orthogonal Idempotent Latin Square of Order 10

摘要

To denote a (3,l,2)-conjugate orthogonal idempotent latin square of order n, the usual acronym is (3,l,2)-COILS(n). Up to now, existence of a (3,l,2)-COILS(n) had been proved for every positive integer n except n = 2,3,4,6, for which the problem was answered in the negative, and n = 10, for which it remained open. In this paper, we use a computer program to prove that a (3,1,2)-COILS(10) does not exist. Following along the lines of recent studies which led to the solution, by means of computer programs, of many open latin square problems, we use a constraint satisfaction technique combining an economical representation of (3,1,2)-COILS with a drastic reduction of the search space. In this way, resolution time is improved by a ratio of 10~4, as compared with current computer programs. Thanks to this improvement in performance, we are able to prove the non-existence of a (3,1,2)-COILS(10).