
COMMENTS

A conference matrix of order n is an n X n {1,0,+1} matrix A such that A A' = (n1)I.
If n == 2 (mod 4) then a necessary condition is that n1 is a sum of 2 squares. It is conjectured that this condition is also sufficient. If n == 2 mod 4 and n1 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.
N. A. Balonin and Jennifer Seberry, A review and new symmetric conference matrices, http://ro.uow.edu.au/cgi/viewcontent.cgi?article=3757&context=eispapers, 2014.
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).


EXTENSIONS

66 seems to be the smallest order for which it is not known if a matrix exists. Since 65 is the sum of two squares, according to the conjecture, 66 should be the next term.
Edited by N. J. A. Sloane, Mar 13 2008, Mar 16 2008, May 22 2014.
