A217332 Orders n of Hadamard cyclic difference sets.

3, 7, 11, 15, 19, 23, 31, 35, 43, 47, 59, 63, 67, 71, 79, 83, 103, 107, 127, 131, 139, 143, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 255, 263, 271, 283, 307, 311, 323, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 511, 523

Offset

1,1

Comments

These are cyclic difference sets (v, k, lambda) with n = k - lambda.

A necessary condition is that n is 3 mod 4, and known sufficient conditions are that n is:

a power of 2 minus 1, or

a prime, or

a product of twin primes.

These sufficient conditions describe all cases below 3439, that is, 3439 is the first number of the form 4k+3 which belongs to none of the three classes above and for which it is not known whether a Hadamard cyclic difference set exists of that order. The known sequence thus extends only as far as 3407.

References

M. Hall, Jr., Combinatorial Theory, 2nd. ed., Wiley, 1986.

M. Harwit and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, 1979. See Appendix.

Links

Veit Elser, Table of n, a(n) for n = 1..255

Leonard D. Baumert, Difference sets, SIAM J. Appl. Math., 17 (1969), 826-833.

Leonard D. Baumert and Daniel M. Gordon, On the existence of cyclic difference sets with small parameters, Proceedings of Conference in Number Theory in Honour of Professor H.C. Williams, 2003.

Example

The first row of the corresponding n X n matrices, from the tables in Harwit and Sloane, 1979 (the other rows are cyclic shifts of the first row):
n=3: 101
n=7: 11101 00
n=11: 11011 10001 0
n=15: 00010 01101 0111
n=19: 11001 11101 01000 0110
n=23: 11111 01011 00110 01010 000

Crossrefs

Sequence in context: A194397 A330213 A039957 * A369056 A079422 A310210

Adjacent sequences: A217329 A217330 A217331 * A217333 A217334 A217335

Keyword

nonn

Author

Veit Elser, Sep 30 2012

Status

approved