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Sequences with small correlation

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Abstract

The extent to which a sequence of finite length differs from a shifted version of itself is measured by its aperiodic autocorrelations. Of particular interest are sequences whose entries are 1 or \(-1\), called binary sequences, and sequences whose entries are complex numbers of unit magnitude, called unimodular sequences. Since the 1950s, there is sustained interest in sequences with small aperiodic autocorrelations relative to the sequence length. One of the main motivations is that a sequence with small aperiodic autocorrelations is intrinsically suited for the separation of signals from noise, and therefore has natural applications in digital communications. This survey reviews the state of knowledge concerning the two central problems in this area: How small can the aperiodic autocorrelations of a binary or a unimodular sequence collectively be and how can we efficiently find the best such sequences? Since the analysis and construction of sequences with small aperiodic autocorrelations is closely tied to the (often much easier) analysis of periodic autocorrelation properties, several fundamental results on corresponding problems in the periodic setting are also reviewed.

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Acknowledgments

I would like to thank Christian Günther, Jonathan Jedwab, Dieter Jungnickel, and Peter Wild for some careful comments on a draft of this survey.

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  1. Department of Mathematics, University of Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Kai-Uwe Schmidt

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This is one of several papers published in Designs, Codes and Cryptography comprising the 25th Anniversary Issue.

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Schmidt, KU. Sequences with small correlation. Des. Codes Cryptogr. 78, 237–267 (2016). https://doi.org/10.1007/s10623-015-0154-7

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