Wieferich Pairs and Barker Sequences, Ii
نویسندگان
چکیده
We show that if a Barker sequence of length n > 13 exists, then either n = 3 979 201 339 721 749 133 016 171 583 224 100, or n > 4 · 1033. This improves the lower bound on the length of a long Barker sequence by a factor of nearly 2000. We also obtain 18 additional integers n < 1050 that cannot be ruled out as the length of a Barker sequence, and find more than 237000 additional candidates n < 10100, the vast majority of which appear likely to satisfy all of the known restrictions on the length of a Barker sequence. These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on n, to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.
منابع مشابه
Wieferich pairs and Barker sequences
We show that if a Barker sequence of length n > 13 exists, then either n = 189 260 468 001 034 441 522 766 781 604, or n > 2 · 1030. This improves the lower bound on the length of a long Barker sequence by a factor of more than 107. We also show that all but fewer than 1600 integers n ≤ 4 ·1026 can be eliminated as the order of a circulant Hadamard matrix. These results are obtained by completi...
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