If there exists a cyclic Hadamard difference set of length v, then v = 4n - 1 is conjectured to be either a prime, or a product of *twin primes*, or one less than a power of 2.
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The main questions are: (1) for what values of v = 4n-1 do these cyclic Hadamard difference sets exist, and (2) what constructions are known to generate them? In Baumert's (1971) book, it is mentioned that all known examples of cyclic Hadamard difference sets have values of v from only three different 'families': (a) v = 4n - 1 is a prime number, (b) v = p(p + 2) is a product of 'twin primes', or (c) v -= U-1, for t = 2,3,4,... It is also reported in Baumert (1971) that there are no other values for v < 1000 with cyclic Hadamard difference sets, with 6 possible exceptions which are v = 399,495,627,651,783, and 975. It turned out that these six cases are also ruled out for the existence of cyclic Hadamard difference sets (Song and Golomb, 1994). In conclusion, there are no counterexamples to the following conjecture for v < 1000: tf there exists a cyclic Hadamard difference set of length v, then * Corresponding author. I Supported in part by the United States Office of Naval Research under Grant No. N00014-90-J-1341.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics