%I M1574 N0615 #53 Jul 25 2023 23:32:24

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

%N Numbers k == 2 (mod 4) that are the orders of conference matrices.

%C A conference matrix of order k is a k X k {-1,0,+1} matrix A such that A A' = (k-1)I.

%C 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.

%D V. Belevitch, Conference matrices and Hadamard matrices, Ann. Soc. Scientifique Bruxelles, 82 (I) (1968), 13-32.

%D CRC Handbook of Combinatorial Designs, 1996, Chapter 52.

%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Joerg Arndt, <a href="/A000952/a000952.txt">Some relevant PARI/GP programs</a>

%H N. A. Balonin and Jennifer Seberry, <a href="https://ro.uow.edu.au/eispapers/2748">A review and new symmetric conference matrices</a>, 2014.

%H Nikolay Balonin, Mikhail Sergeev and Anton Vostrikov, <a href="https://doi.org/10.31799/1684-8853-2020-2-2-9">Prime Fermat numbers and maximum determinant matrix conjecture</a>, Information and Control Systems (2020) No. 2, 2-9. (Abstract in Russian, English translation available on page)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Conference_matrix">Conference matrix</a>.

%e The essentially unique conference matrix of order 6:

%e 0 +1 +1 +1 +1 +1

%e +1 0 +1 -1 -1 +1

%e +1 +1 0 +1 -1 -1

%e +1 -1 +1 0 +1 -1

%e +1 -1 -1 +1 0 +1

%e +1 +1 -1 -1 +1 0

%Y Subsequence of A016825.

%Y Cf. A286636.

%K nonn,hard,more,nice

%O 1,1

%A _N. J. A. Sloane_

%E 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.

%E Edited by _N. J. A. Sloane_, Mar 13 2008, Mar 16 2008, May 22 2014