The non-existence of (3,1,2)-conjugate orthogonal idemôtent latin square of order 10

Olivier Dubois (1) , Gilles Dequen

1 SYSDEF - Systèmes d'aide à la décision et à la formation

To denote a (3,1,2)-conjugate orthogonal idempotent latin square of order n, the usual acronym is (3,1,2)-COILS(n). Up to now, existence of a (3,1,2)-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 104, 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).

Domaines: Informatique [cs]

Olivier Dubois, Gilles Dequen. The non-existence of (3,1,2)-conjugate orthogonal idemôtent latin square of order 10. CP 2001 - 7th International Conference on Principles and Practice of Constraint Programming, Nov 2001, Paphos, Cyprus. pp.108-120, ⟨10.1007/3-540-45578-7_8⟩. ⟨hal-01571006⟩