Lam 1997 search finite projective plane order 10

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main_theorem: Lam's account that the computer searches for codewords of weights 12, 16 and 19 in the binary code of a putative plane of order 10 completed without finding a plane; the article reports the result as an experimental one and proves none of it.

theorem_2: Lam's statement of Bose's theorem that, for n at least 3, a finite projective plane of order n exists exactly when a complete set of n - 1 mutually orthogonal Latin squares of order n exists; the article cites it and does not prove it.

theorem_3: Lam's statement of the Bruck--Ryser theorem: if n is congruent to 1 or 2 modulo 4 and a finite projective plane of order n exists, then n is a sum of two integer squares; the article cites it and does not prove it.

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C. W. H. Lam, The search for a finite projective plane of order 10, Amer. Math. Monthly 98 (1991), no. 4, 305--318; MR1103185 (92b:51013). The citing page's entry [La97], "(1997), 335-355", is a reprint of the Monthly article; the reprint volume was not consulted and was not identified from the copy read.

The copy read for this card is the author's TeX revision of the article dated November 30, 2005 (dvipdfm output, 23 A4 pages, complete text layer). Neither the 1991 printing nor the 1997 reprint was consulted, and the text read was not compared with either; its reference [21] still lists the Lam, Thiel and Swiercz nonexistence paper as "to appear", so the reference list was not updated to that paper's 1989 publication. Page references are to that revision's own page numbers. Provenance: the copy read came from a survey download of September 2026; the download URL was not recorded; 151,808 bytes. That copy is the author's TeX revision dated November 30, 2005, which prints no copyright or license line on pp. 1--2 or 22--23; its download URL was not recorded, so no host's terms could be checked, and the published version is not the edition read, so no publisher page applies; the term is unstated.

Read status: claims checked for the theorems and the account of the search listed below (read clause by clause on the text layer); the article is expository and proves none of the nonexistence results it reports.

Contents

• Section 2 (pp. 1--8): definition of a finite projective plane of order $n$ ($n^2+n+1$ points and lines, $n+1$ points per line, $n+1$ lines per point, unique meets and joins); the plane of order 2 (Fig. 1); the small planes of Veblen, Bussey and Wedderburn; Bose's 1938 explanation of the missing order 6. Theorem 1 (p. 4): $t$ mutually orthogonal Latin squares of order $n\ge3$ satisfy $t\le n-1$. Theorem 2 (Bose; p. 4): a projective plane of order $n\ge3$ exists iff a complete set of $n-1$ mutually orthogonal Latin squares of order $n$ exists. Tarry's enumeration (around 1900) settles order 6 (p. 5).
• Theorem 3 (Bruck and Ryser; p. 5): if $n\equiv1,2\pmod4$ and a plane of order $n$ exists, then $n=x^2+y^2$ for integers $x,y$. The article does not repeat the proof; pp. 6--7 explain only its starting point, the incidence-matrix equation $AA^T=nI+J$. Theorem 4 (Hall and Ryser; p. 7) is the partial converse with rational matrices; the Bruck--Ryser--Chowla extension to symmetric designs is mentioned (p. 7). Order 10 passes the test since $10=1^2+3^2$ (p. 7).
• Sections 3--4 (pp. 8--18): the coding-theory approach of Assmus and Mattson and of MacWilliams, Sloane and Thompson, the binary code of a plane of order 10 and its weight enumerator; Theorem 5 (p. 9): $|v\cap l|\equiv|v|\pmod2$ for every line $l$ and codeword $v$; the cases of weights 12, 15, 16 and 19, the earlier computer results, and the design of the weight-19 search.
• Section 5 (pp. 18--19): the CRAY-1A run was reported finished on November 11, 1988; two of its cases (A2's) had given error number 4, a size problem for a data structure that could not be enlarged, and each was completed with a modified CRAY program and the NPL program, the first by November 29, 1988 and the second by the end of January 1989, when "the plane of order 10 was dead a third and hopefully the final time" (p. 19). The completed search found no projective plane of order 10.
• Section 6 (pp. 19--20): the standing of a computer proof; the estimated probability that undetected hardware errors hide a plane; "the fact that no one has yet constructed one is a very strong indication that it does not exist" (p. 20).

Result pages

• Theorem 2 (p. 4): Bose's theorem, for $n\ge3$, that a plane of order $n$ exists iff a complete set of $n-1$ mutually orthogonal Latin squares of order $n$ does; with Theorem 1 (p. 4) and Tarry's order-6 enumeration (p. 5).
• Theorem 3 (p. 5): the Bruck--Ryser theorem, with the partial converse Theorem 4 (p. 7).
• Reported result (pp. 8--19): the chain of computer searches, for codewords of weights 15, 12, 16 and 19, that found no plane of order 10.

Theorems 1, 4 and 5 are recorded on those pages and not given pages of their own.

Compiled scope

The whole text was read once for its statements; none of the reported results (Bruck--Ryser, the code-theoretic lemmas, the search itself) is verified here, and the printed versions were not compared. Nothing here is independently reviewed.

Bears on. #723, whether every finite projective plane has prime-power order:

• the reported result excludes order 10, which is not a prime power and passes the Bruck--Ryser test since $10=1^2+3^2$; the article reports the computer search and proves none of it.
• Theorem 3 (Bruck--Ryser, cited, not proved here) excludes every order $n\equiv1,2\pmod4$ that is not a sum of two squares, among them 6.
• Theorem 2 (Bose, cited, not proved here) excludes order 6 together with Tarry's enumeration, which the article reports.

None of them decides order 12, an order $\equiv0\pmod4$ that is not a prime power.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.