โ Scribed by C.W.H. Lam; G. Kolesova; L. Thiel
No coin nor oath required. For personal study only.
Lam, C.W.H., G. Kolesova and L. Thiel, A computer search for iinite projective planes of order 9, Discrete Mathematics 92 (1991) 187-195. There 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 fmther 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 iatin squares. 1. Ilttroduction A finite projective plane of order n is a collection of n2 + n + 1 lines and n2 + n + 1 points such that: (1) every line contains n + 1 points, (2) every point is on n + 1 lines, (3) any two distinct lines intersect at exactly one point, and (4) any two distinct points lie on exactly one iine. There are four known planes of order 9, namely the Desarguesian Plane, the Left Nearfield Plane, the Right Nearfield Pane and the Hughes Plane. For a detailed description of these planes, as well as their collineation groups, see [ll]. *This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grants A9373, A9413, 0011 and by the Fonds pour la Formation de Chersbeurs et I'Aide ii la Recherche under Grants EQ2369 and EQ3886.
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