A binarization of geometric sequences with Legendre symbol and its autocorrelation

Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let ω be its zero, that is a primitive element in F_pm*, then the sequence S = {s_i}, s_i = Tr (ωⁱ) for i = 0, 1, 2, ... becomes a maximum length sequence, where Tr (·) is the trace function over F_p. 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^m - 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.