A binarization of geometric sequences with Legendre symbol and its autocorrelation

Abstract

Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let &#x03C9; be its zero, that is a primitive element in F<sub>pm</sub>*, then the sequence S = {s<sub>i</sub>}, s<sub>i</sub> = Tr (&#x03C9;<sup>i</sup>) for i = 0, 1, 2, ... becomes a maximum length sequence, where Tr (&#x00B7;) is the trace function over F<sub>p</sub>. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period and autocorrelation has not been discussed. Then, it is shown that the obtained binary sequence (geometric sequence with Legendre symbol) has the period L given by 2(p<sup>m</sup> - 1)/(p-1) and a certain periodic autocorrelation. After that, this paper also shows the numbers of ones and minus ones in the proposed binary sequence per a period together with some small examples.

DOI: 10.1109/IWSDA.2013.6849054

Cite this paper

@article{Nogami2013ABO, title={A binarization of geometric sequences with Legendre symbol and its autocorrelation}, author={Yasuyuki Nogami and Kazuki Tada and Satoshi Uehara}, journal={The Sixth International Workshop on Signal Design and Its Applications in Communications}, year={2013}, pages={28-31} }