A geometric sequence binarized with legendre symbol over odd characteristic field and its properties

Let p be an odd characteristic and m be the degree of a primitive polynomial f (x) over the prime field F_p. Let ω be its zero, that is a primitive element in F^∗ _pm , the sequence S = {si}, si = Tr (ωⁱ) for i = 0, 1, 2, ⋯ becomes a non-binary 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, autocorrelation, and linear complexity have not been discussed. Then, this paper shows that the generated binary sequence (geometric sequence by Legendre symbol) has the period n = 2(p^m ? 1)/(p ? 1) and a typical periodic autocorrelation. Moreover, it is experimentally observed that its linear complexity becomes the maximum, that is the period n. Among such experimental observations, especially in the case of m = 2, it is shown that the maximum linear complexity is theoretically proven. After that, this paper also demonstrates these properties with a small example.