A computer search for finite projective planes of order 9

AbstractThere are four known finite projective planes of order 9. This paper reports the result of a computer search which shows that this list is complete. The computer search starts by generating all 283,657 non-isomorphic latin squares of order 8. Each latin square gives 27 columns of the incidence matrix. Another program attempts to complete each of these incidence matrices to 40 columns. Only 21 of them can be so completed, giving rise to 326 matrices of 40 columns. A third computer program attempts to complete the rest of the matrices. One of the 326 does not complete. The rest complete each to a unique matrix. An isomorphism testing program is then applied to the 325 complete matrices, creating a certificate for each matrix, as well as its collineation group. The certificates are then compared with the known planes and no new ones found. As a further evidence of the correctness of the computer programs, this paper also shows that the computer results are consistent with those expected by using information about the known planes and their associated latin squares