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Claim. No finite projective plane of order $10$ exists. The result completes a chain of computer searches. A plane of order $10$ would give a binary code of length $111$ whose weight enumerator is fixed by its numbers of codewords of weights $12$, $15$ and $16$ (Assmus and Mattson). Weight $15$ was excluded by MacWilliams, Sloane and Thompson (J. Combin. Theory Ser. A 14 (1973), 66–78), weight $12$ by Lam, Thiel, Swiercz and McKay (Discrete Math. 45 (1983), 319–321) and weight $16$ by Lam, Thiel and Swiercz (J. Combin. Theory Ser. A 42 (1986), 207–214). With those three numbers zero the enumerator requires 24,675 codewords of weight $19$, and the search this paper reports shows that no codeword of weight $19$ can be completed to a plane, the CRAY-1A run finishing on 11 November 1988, with two subcases rerun by 29 November 1988 and the end of January 1989, after about 2,000 hours of CRAY time by the later estimate [La97] reports (the authors' own guess was 3,000). Order $10$ is the smallest order that is not a prime power and passes the Bruck–Ryser test, since $10=1^2+3^2$, so the search is the first exclusion beyond Bruck and Ryser's theorem. The site credits the exclusion to the computer search through Lam's expository account [La97], recorded on the library's card, which reports the search's history and design and proves none of the results itself.

Covers. The order $n=10$: no plane of that order exists, so the implication of Problem 723 holds for $n=10$. Together with Bruck and Ryser's exclusion of $n=6$ the conjecture holds for every $n\le11$, and $n=12$ is the first order it leaves undecided.

Acceptance. Refereed: C. W. H. Lam, L. Thiel and S. Swiercz, The non-existence of finite projective planes of order 10, Canad. J. Math. 41 (1989), no. 6, 1117–1123; the issue is dated December 1989, and the page is dated to the first day of that month. The site's curator records the exclusion under [La97], Lam's Monthly article of 1991, while labeling the problem FALSIFIABLE, which credits the partial result without settling the problem, so the page lists no reviewed evidence. The search was not repeated and the paper is not held in this corpus; the account above follows Lam's expository article [La97].