A000952 Numbers k == 2 (mod 4) that are the orders of conference matrices.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

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

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.

Examples

			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
		

References

• 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).

Links

• Joerg Arndt, Some relevant PARI/GP programs
• N. A. Balonin and Jennifer Seberry, A review and new symmetric conference matrices, 2014.
• Nikolay Balonin, Mikhail Sergeev and Anton Vostrikov, Prime Fermat numbers and maximum determinant matrix conjecture, Information and Control Systems (2020) No. 2, 2-9. (Abstract in Russian, English translation available on page)
• Wikipedia, Conference matrix.

Crossrefs

Subsequence of A016825.

Cf. A286636.

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.

Edited by N. J. A. Sloane, Mar 13 2008, Mar 16 2008, May 22 2014