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A217332
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Orders n of Hadamard cyclic difference sets.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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M. Hall, Jr., Combinatorial Theory, 2nd. ed., Wiley, 1986.
M. Harwit and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, 1979. See Appendix.
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LINKS
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Leonard D. Baumert, Difference sets, SIAM J. Appl. Math., 17 (1969), 826-833.
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EXAMPLE
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Veit Elser, Sep 30 2012
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STATUS
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approved
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