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A000952
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Numbers k == 2 (mod 4) that are the orders of conference matrices.
(Formerly M1574 N0615)
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8
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2, 6, 10, 14, 18, 26, 30, 38, 42, 46, 50, 54, 62
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OFFSET
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1,1
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COMMENTS
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A conference matrix of order k is a k X k {-1,0,+1} matrix A such that A A' = (k-1)I.
If k == 2 (mod 4) then a necessary condition is that k-1 is a sum of 2 squares (A286636). It is conjectured that this condition is also sufficient. If k == 2 (mod 4) and k-1 is a prime or prime power the condition is automatically satisfied.
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REFERENCES
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V. Belevitch, Conference matrices and Hadamard matrices, Ann. Soc. Scientifique Bruxelles, 82 (I) (1968), 13-32.
CRC Handbook of Combinatorial Designs, 1996, Chapter 52.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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The essentially unique conference matrix of order 6:
0 +1 +1 +1 +1 +1
+1 0 +1 -1 -1 +1
+1 +1 0 +1 -1 -1
+1 -1 +1 0 +1 -1
+1 -1 -1 +1 0 +1
+1 +1 -1 -1 +1 0
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CROSSREFS
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KEYWORD
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nonn,hard,more,nice
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AUTHOR
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EXTENSIONS
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66 seems to be the smallest order for which it is not known whether a conference matrix exists. Since 65 is the sum of two squares, according to the conjecture, 66 should be the next term.
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STATUS
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approved
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