Designs, Codes and Cryptography

, Volume 53, Issue 3, pp 149–163 | Cite as

Wieferich pairs and Barker sequences

Article

Abstract

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 completing extensive searches for Wieferich prime pairs (q, p), which are defined by the relation \({q^{p-1} \equiv1}\) mod p2, and analyzing their results in combination with a number of arithmetic restrictions on n.

Keywords

Barker sequence Circulant Hadamard matrix Wieferich prime pair 

Mathematics Subject Classifications (2000)

11B85 05B10 11A41 94A55 

References

  1. 1.
    Barker R.H.: Group synchronizing of binary digital systems. In: Jackson, W. (eds) Communication Theory (London, 1952), pp. 273–287. Academic Press, New York (1953)Google Scholar
  2. 2.
    Baumert L.D.: Cyclic Difference Sets. Lecture Notes in Math., vol. 182. Springer-Verlag (1971) (MR0282863 (44 #97)).Google Scholar
  3. 3.
    Borwein P., Kaltofen E., Mossinghoff M.J.: Irreducible polynomials and barker sequences. ACM Commun. Comput. Algebra 41(3–4), 118–121 (2007) (MR2404490)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Borwein P., Mossinghoff M.J.: Barker sequences and flat polynomials. In: McKee J., Smyth C. (eds.) Number Theory and Polynomials (Bristol, U.K., 2006). London Math. Soc. Lecture Note Ser., vol. 352, pp. 71–88. Cambridge Univ. Press (2008).Google Scholar
  5. 5.
    Brualdi R.A.: A note on multipliers of difference sets. J. Res. Nat. Bur. Stand. Sect. B 69, 87–89 (1965) (MR0184868 (32 #2339))MATHMathSciNetGoogle Scholar
  6. 6.
    Crandall R., Dilcher K., Pomerance C.: A search for Wieferich and Wilson primes. Math. Comp. 66(217), 433–449 (1997) (MR1372002 (97c:11004))MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Eliahou S., Kervaire M.: Barker sequences and difference sets. Enseign. Math. (2) 38(3–4), 345–382 (1992). Corrigendum, Enseign. Math. (2) 40(1–2), 109–111 (1994) (MR1189012 (93i:11018)).Google Scholar
  8. 8.
    Eliahou S., Kervaire M., Saffari B.: A new restriction on the lengths of Golay complementary sequences. J. Combin. Theory Ser. A 55(1), 49–59 (1990) (MR1070014 (91i:11020))MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ernvall R., Metsänkylä T.: On the p-divisibility of Fermat quotients. Math. Comp. 66(219), 1353–1365 (1997) (MR1408373 (97i:11003))MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fischer R.: Fermat quotients, http://www.fermatquotient.com/FermatQuotienten.
  11. 11.
    Fredman M.L., Saffari B., Smith B.: Polynômes réciproques: conjecture d’Erdős en norme L 4, taille des autocorrélations et inexistence des codes de Barker. C. R. Acad. Sci. Paris Sér. I Math. 308(15), 461–464 (1989) (MR994692 (90c:42004))MATHMathSciNetGoogle Scholar
  12. 12.
    GMP: The GNU multiple precision arithmetic library, http://www.swox.com/gmp.
  13. 13.
    Granville A., Monagan M.B.: The first case of Fermat’s last theorem is true for all prime exponents up to 714,591,416,091,389. Trans. Am. Math. Soc. 306(1), 329–359 (1988) (MR0927694 (89g:11025))MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jedwab J., Lloyd S.: A note on the nonexistence of Barker sequences. Des. Codes Cryptogr. 2(1), 93–97 (1992) (MR1157481 (93e:11032))MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Keller W., Richstein J.: Solutions of the congruence \({a^ {p-1} \equiv1\pmod{p^ r}}\). Math. Comp. 74(250), 927–936 (2005) (MR2114655 (2005i:11004))MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Knauer J., Richstein J.: The continuing search for Wieferich primes. Math. Comp. 74(251), 1559–1563 (2005) (MR2137018 (2006a:11006))MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lehmer D.H.: On Fermat’s quotient, base two. Math. Comp. 36(153), 289–290 (1981) (MR595064 (82e:10004))MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Leung K.H., Schmidt B.: The field descent method. Des. Codes Cryptogr. 36(2), 171–188 (2005) (MR2211106 (2007g:05023))MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mihăilescu P.: Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004) (MR2076124 (2005f:11051))MATHMathSciNetGoogle Scholar
  20. 20.
    Mihăilescu P.: A class number free criterion for Catalan’s conjecture. J. Number Theory 99(2), 225–231 (2003) (MR1968450 (2004b:11040))MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Montgomery P.L.: New solutions of \({a^ {p-1} \equiv 1\pmod {p^2}}\). Math. Comp. 61(203), 361–363 (1993) (MR1182246 (94d:11003))MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Mossinghoff M.J.: Wieferich prime pairs, Barker sequences, and circulant Hadamard matrices (2009). http://www.cecm.sfu.ca/~mjm/WieferichBarker.
  23. 23.
    Nielsen P.P.: Odd perfect numbers have at least nine distinct prime factors. Math. Comp. 76(260), 2109–2126 (2007)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Ryser H.J.: Combinatorial Mathematics. Carus Math. Monogr., vol. 14. Math. Assoc. Am. (1963) (MR0150048 (27 #51)).Google Scholar
  25. 25.
    Schmidt B.: Cyclotomic integers and finite geometry. J. Amer. Math. Soc. 12(4), 929–952 (1999) (MR1671453 (2000a:05042))MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Schmidt B.: Characters and Cyclotomic Fields in Finite Geometry. Lecture Notes in Math., vol. 1797. Springer-Verlag (2002) (MR1943360 (2004a:05028)).Google Scholar
  27. 27.
    Tarjan R.: Enumeration of the elementary circuits of a directed graph. SIAM J. Comput. 2, 211–216 (1973) (MR0325448 (48 #3795))MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Turyn R.: Character sums and difference sets. Pacific J. Math. 15, 319–346 (1965) (MR0179098 (31 #3349))MATHMathSciNetGoogle Scholar
  29. 29.
    Turyn R.: Sequences with small correlation. In: Mann, H.B. (eds) Error Correcting Codes (Madison, WI, 1968), pp. 195–228. Wiley, New York (1968) (MR0242566 (39 #3897))Google Scholar
  30. 30.
    Turyn R., Storer J.: On binary sequences. Proc. Am. Math. Soc. 12, 394–399 (1961) (MR0125026 (23 #A2333))MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wieferich A.: Zum letzten Fermat’schen theorem. J. Reine Angew. Math. 136, 293–302 (1909)MATHGoogle Scholar
  32. 32.
    Wieferich@Home (2009). http://www.elmath.org.

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsDavidson CollegeDavidsonUSA

Personalised recommendations