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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a class of polynomials related to Barker sequences


Authors: Peter Borwein, Stephen Choi and Jonas Jankauskas
Journal: Proc. Amer. Math. Soc. 140 (2012), 2613-2625
MSC (2010): Primary 11B83, 11C08, 30C10; Secondary 42A05, 94A05
Published electronically: December 8, 2011
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Abstract: For an odd integer $ n > 0$, we introduce the class $ \mathcal {L}P_n$ of Laurent polynomials

$\displaystyle P(z) = (n+1) + \sum _{\substack {k = 1 \\ k \text { odd}}}^{n}c_k (z^k+z^{-k}), $

with all coefficients $ c_k$ equal to $ -1$ or $ 1$. Such polynomials arise in the study of Barker sequences of even length, i.e., integer sequences having minimal possible autocorrelations. We prove that polynomials $ P \in \mathcal {L}P_n$ have large Mahler measures, namely, $ M(P) > (n+1)/2$. We conjecture that minimal Mahler measures in the class $ \mathcal {L}P_n$ are attained by the polynomials $ R_n(z)$ and $ R_n(-z)$, where

$\displaystyle R_n(z) = (n+1) + \sum _{\substack {k = -n \\ k \text { odd}}}^{n} z^k $

is a polynomial with all the coefficients $ c_k=1$. We prove that

$\displaystyle M(R_n) > n - \frac {2}{\pi } \log {n} + O(1). $

The results of experimental computations on polynomials in the class $ \mathcal {L}P_n$ suggest two conjectures which could shed light on the long-standing Barker problem.

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Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada
Email: pborwein@sfu.ca

Stephen Choi
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada
Email: kkchoi@math.sfu.ca

Jonas Jankauskas
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Email: jonas.jankauskas@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11114-6
PII: S 0002-9939(2011)11114-6
Keywords: Laurent polynomials, Barker conjecture, Barker polynomials, Barker sequences, Littlewood polynomials, Mahler measures, $L_{p}$ norms, aperiodic autocorrelations
Received by editor(s): January 23, 2011
Received by editor(s) in revised form: March 6, 2011
Published electronically: December 8, 2011
Additional Notes: The first and second authors are supported by NSERC, Canada.
A visit of the third author at IRMACS Center, Simon Fraser University, was funded by the Lithuanian Research Council (student research support project).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.