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ON THE GOWERS NORM OF PSEUDORANDOM BINARY SEQUENCES

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Exponential sums and character sums

Published online by Cambridge University Press:  13 March 2009

HARALD NIEDERREITER  and
JOËL RIVAT
HARALD NIEDERREITER
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore (email: nied@math.nus.edu.sg)
JOËL RIVAT*
Affiliation:
Institut de Mathématiques de Luminy, CNRS-UMR 6206, Université de la Méditerranée, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: rivat@iml.univ-mrs.fr)
*
✉For correspondence; e-mail: rivat@iml.univ-mrs.fr
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Abstract

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We study the Gowers norm for periodic binary sequences and relate it to correlation measures for such sequences. The case of periodic binary sequences derived from inversive pseudorandom numbers is considered in detail.

Keywords

Gowers normpseudorandombinary sequenceadditive charactercorrelation

MSC classification

Secondary: 11K45: Pseudo-random numbers; Monte Carlo methods 11L40: Estimates on character sums
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 259 - 271
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Goubin, L., Mauduit, Ch. and Sárközy, A., ‘Construction of large families of pseudorandom binary sequences’, J. Number Theory 106 (2004), 56–69.CrossRefGoogle Scholar
[2] Gowers, W. T., ‘A new proof of Szemerédi’s theorem’, Geom. Funct. Anal. 11 (2001), 465–588.Google Scholar
[3] Mauduit, Ch., Niederreiter, H. and Sárközy, A., ‘On pseudorandom [0, 1) and binary sequences’, Publ. Math. Debrecen 71 (2007), 305–324.CrossRefGoogle Scholar
[4] Mauduit, Ch. and Sárközy, A., ‘On finite pseudorandom binary sequences I: Measure of pseudorandomness, the Legendre symbol’, Acta Arith. 82 (1997), 365–377.CrossRefGoogle Scholar
[5] Moreno, C. J. and Moreno, O., ‘Exponential sums and Goppa codes: I’, Proc. Amer. Math. Soc. 111 (1991), 523–531.Google Scholar
[6] Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods (SIAM, Philadelphia, PA, 1992).CrossRefGoogle Scholar
[7] Niederreiter, H. and Rivat, J., ‘On the correlation of pseudorandom numbers generated by inversive methods’, Monatsh. Math. 153 (2008), 251–264.CrossRefGoogle Scholar
[8] Niederreiter, H., Rivat, J. and Sárközy, A., ‘Pseudorandom sequences of binary vectors’, Acta Arith. 133 (2008), 109–125.CrossRefGoogle Scholar
[9] Niederreiter, H. and Shparlinski, I. E., ‘Recent advances in the theory of nonlinear pseudorandom number generators’, in: Monte Carlo and Quasi-Monte Carlo Methods 2000 (Springer, Berlin, 2000), pp. 86–102.Google Scholar
[10] Niederreiter, H. and Winterhof, A., ‘Incomplete exponential sums over finite fields and their applications to new inversive pseudorandom number generators’, Acta Arith. 93 (2000), 387–399.CrossRefGoogle Scholar
[11] Niederreiter, H. and Winterhof, A., ‘On the structure of inversive pseudorandom number generators’, in: AAECC 2007, Lecture Notes in Computer Science, 4851 (Springer, Berlin, 2007), pp. 208–216.Google Scholar