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International Workshop on the Arithmetic of Finite Fields

WAIFI 2018: Arithmetic of Finite Fields pp 149–172Cite as

Sequences with Low Correlation

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11321))

Abstract

Pseudorandom sequences are used extensively in communications and remote sensing. Correlation provides one measure of pseudorandomness, and low correlation is an important factor determining the performance of digital sequences in applications. We consider the problem of constructing pairs (f, g) of sequences such that both f and g have low mean square autocorrelation and f and g have low mean square mutual crosscorrelation. We focus on aperiodic correlation of binary sequences, and review recent contributions along with some historical context.

This paper is based upon work supported in part by the National Science Foundation under Grants DMS-1500856 and CCF-1815487.

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Notes

  1. 1.

    This sequence was formed using the specific choice of 1 as the additive generator of \({\mathbb F}_p\). We could have replaced 1 with any other \(a\in {\mathbb F}_p^{*}\) to form the list \(0,a,2a,\ldots ,(p-1)a\) of elements of \({\mathbb F}_p\) instead of (13), and then apply \(\chi \) to every term to get \(\left( \chi (0),\chi (a),\chi (2 a),\ldots ,\chi ((p-1)a)\right) \). This would just give the sequence in (14) multiplied by the unimodular scalar \(\chi (a)\). This scalar multiplciation has no effect on the magnitudes of correlation values.

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Acknowledgement

The author thanks Yakov Sapozhnikov for his careful reading of this paper and his helpful suggestions.

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  1. Department of Mathematics, California State University, Northridge, CA, 91330, USA

    Daniel J. Katz

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  1. Daniel J. Katz
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Correspondence to Daniel J. Katz .

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  1. Department of Informatics, University of Bergen, Bergen, Norway

    Lilya Budaghyan

  2. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico, Mexico

    Francisco Rodríguez-Henríquez

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Katz, D.J. (2018). Sequences with Low Correlation. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_8

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  • DOI: https://doi.org/10.1007/978-3-030-05153-2_8

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