

A000952


Numbers k == 2 (mod 4) that are the orders of conference matrices.
(Formerly M1574 N0615)


8



2, 6, 10, 14, 18, 26, 30, 38, 42, 46, 50, 54, 62
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OFFSET

1,1


COMMENTS

A conference matrix of order k is a k X k {1,0,+1} matrix A such that A A' = (k1)I.
If k == 2 (mod 4) then a necessary condition is that k1 is a sum of 2 squares (A286636). It is conjectured that this condition is also sufficient. If k == 2 (mod 4) and k1 is a prime or prime power the condition is automatically satisfied.


REFERENCES

V. Belevitch, Conference matrices and Hadamard matrices, Ann. Soc. Scientifique Bruxelles, 82 (I) (1968), 1332.
CRC Handbook of Combinatorial Designs, 1996, Chapter 52.
F. J. MacWilliams and N. J. A. Sloane, The Theory of ErrorCorrecting Codes, ElsevierNorth 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).


LINKS



EXAMPLE

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


CROSSREFS



KEYWORD

nonn,hard,more,nice


AUTHOR



EXTENSIONS

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.


STATUS

approved



