Correlation distribution of a sequence family generalizing some sequences of Trachtenberg

Ferruh Özbudak; Eda Tekin

doi:10.3934/amc.2020087

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2021, Volume 15, Issue 4: 647-662. Doi: 10.3934/amc.2020087

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Correlation distribution of a sequence family generalizing some sequences of Trachtenberg

Ferruh Özbudak^1, and
Eda Tekin^2,3, ,

1.
Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey
2.
Department of Business Administration, Karabük University, 78050, Karabük, Turkey
3.
Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

^* Corresponding author: Eda Tekin
^* Corresponding author: Eda Tekin

Received: November 2019

Early access: June 2020

Published: November 2021

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Abstract

In this paper, we give a classification of a sequence family, over arbitrary characteristic, adding linear trace terms to the function $ g(x) = \mathrm{Tr}(x^d) $, where $ d = p^{2k}-p^k+1 $, first introduced by Trachtenberg. The family has $ p^n+1 $ cyclically distinct sequences with period $ p^n-1 $. We compute the exact correlation distribution of the function $ g(x) $ with linear $ m $-sequences and amongst themselves. The cross-correlation values are obtained as $ C_{i,j}(\tau) \in \{-1,-1\pm p^{\frac{n+e}{2}},-1+p^n\} $.

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Mathematics Subject Classification: Primary: 94A55, 11T71; Secondary: 14G50.

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