<abstract xmlns="http://www.w3.org/1999/xhtml">We exhibit a class of Littlewood polynomials that are not <italic>L</italic><italic>α</italic>-flat for any <italic>α</italic> ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not <italic>L</italic><italic>α</italic>-flat, <italic>α</italic> ≥ 0, when the frequency of −1 is not in the interval ]<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math><tex-math>{1 \over 4}</tex-math></alternatives></inline-formula>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0003_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math><tex-math>{3 \over 4}</tex-math></alternatives></inline-formula>[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not <italic>L</italic><italic>α</italic>-flat for any <italic>α></italic> 2 if the frequency of <italic>−</italic>1 is not <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0003_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math><tex-math>{1 \over 2}</tex-math></alternatives></inline-formula>. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not <italic>L</italic><italic>α</italic>-flat for any <italic>α ≥</italic> 0, and we provide a lemma on the existence of <italic>c</italic>-flat polynomials.</abstract>

We exhibit a class of Littlewood polynomials that are not Lα-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not Lα-flat, α ≥ 0, when the frequency of −1 is not in the interval ]14{1 \over 4}, 34{3 \over 4}[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not Lα-flat for any α> 2 if the frequency of −1 is not 12{1 \over 2}. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not Lα-flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.

A Class of Littlewood Polynomials that are Not <italic>L</italic><italic>α</italic>-Flat

Uniform distribution theory

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A Class of Littlewood Polynomials that are Not -Flat

We exhibit a class of Littlewood polynomials that are not Lα-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not...

A Class of Littlewood Polynomials that are Not L^α-Flat

Published Online: Jul 24, 2020

Page range: 51 - 74

Received: Sep 17, 2019

Accepted: Mar 26, 2020

DOI: https://doi.org/10.2478/udt-2020-0003

Keywords
Merit factor, flat polynomials, ultraflat polynomials, Erd˝ os-Newman flatness problem, Littlewood flatness problem, digital transmission, palindromic polynomial, TurynGolay’s conjecture

© 2020 El Houcein El Abdalaoui et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A Class of Littlewood Polynomials that are Not Lα-Flat

Published Online: Jul 24, 2020

Page range: 51 - 74

Received: Sep 17, 2019

Accepted: Mar 26, 2020

DOI: https://doi.org/10.2478/udt-2020-0003

KeywordsMerit factor, flat polynomials, ultraflat polynomials, Erd˝ os-Newman flatness problem, Littlewood flatness problem, digital transmission, palindromic polynomial, TurynGolay’s conjecture

© 2020 El Houcein El Abdalaoui et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A Class of Littlewood Polynomials that are Not L^α-Flat

Keywords
Merit factor, flat polynomials, ultraflat polynomials, Erd˝ os-Newman flatness problem, Littlewood flatness problem, digital transmission, palindromic polynomial, TurynGolay’s conjecture