In finite geometry, Lam's problem is the problem of determining if a finite projective plane of order ten exists.
The order ten case is the first theoretically uncertain case, as all smaller orders can be resolved by purely theoretical means.
[1] Lam's problem is named after Clement W. H. Lam who experimentally determined that projective planes of order ten do not exist via exhaustive computational searches.
is a collection of points and lines such that A consequence of this definition is that a projective plane of order
Using the incidence matrix representation, Lam's problem is equivalent to determining if there is a way of placing 0s and 1s in a
[3] Lam considered studying the existence of a projective plane of order ten in his PhD thesis but was dissuaded by his thesis advisor H. J. Ryser who believed the problem was too difficult.
[2] Edward Assmus presented a connection between projective planes and coding theory at the conference Combinatorial Aspects of Finite Geometries in 1970.
[4] He studied the code generated by the rows of the incidence matrix of a hypothetical projective plane of order ten and derived a number of restrictive properties that such a code must satisfy.
Over the next two decades a number of computer searches showed that the hypothetical code associated with the projective plane of order ten does not contain words of weights 15,[5] 12,[6] and 16[7]—which implied that it must contain words of weight 19.
Finally, Clement Lam, Larry Thiel and Stanley Swiercz used about three months of time on a Cray-1A supercomputer to show that words of weight 19 are also not present in the code.
Their result was independently verified in 2021 by using a SAT solver to generate computer-verifiable certificates for the correctness of the exhaustive searches.