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
KEYWORD
nonn
AUTHOR
Veit Elser, Sep 30 2012
STATUS
approved