1,1

A conference matrix of order n is an n X n {-1,0,+1} matrix A such that A A' = (n-1)I.

If n == 2 (mod 4) then a necessary condition is that n-1 is a sum of 2 squares. It is conjectured that this condition is also sufficient. If n == 2 mod 4 and n-1 is a prime or prime power the condition is automatically satisfied.

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).

Joerg Arndt, Some relevant gp programs

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

Sequence in context: A122905 A132417 A103747 this_sequence A164302 A039956 A118369

Adjacent sequences: A000949 A000950 A000951 this_sequence A000953 A000954 A000955

nonn,hard,nice

N. J. A. Sloane (njas(AT)research.att.com).

66 seems to be the smallest order for which it is not known if a matrix exists.

Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 13 2008, Mar 16 2008