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.

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 A039956 A118369 A082816

Adjacent sequences: A000949 A000950 A000951 this_sequence A000953 A000954 A000955

nonn,hard,nice

njas

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

Edited by njas, Mar 13 2008, Mar 16 2008