K-Autocorrelation and Its applications

Abstract

The standart autocorrelation is used to measure the similarities between a binary sequence and its any shifted form. It has applications in communication systems and cryptography. Let a =(a_0,a_1,a_2,...,a_n-1) be a binary sequence and its shifted forms be a_tau =(a_-tau,a_1-tau,a_2-tau,...,a_n-1-tau) for tau=1,2,...,n-1, where indices of this sequence are in modulo . The standart autocorrelation of the sequences and is defined by c_tau(a)=sum_i=0^n-1(-1)^a_i+a_i-tau. {c_tau(a)}_tau=0^n-1 sequence is called autocorrelation coefficients. In this study, we define k-autocorrelation of for a binary sequence and its k-1 shifted forms. The k-autocorrelation is a generalization of standart autocorrelation. If we take k=2, then we get the standart autocorrelation. Also, for given tau_1,tau_2,...,tau_k-1 in Z such that 1<=tau_1<tau_2<...<tau_k-1<=n-1, we call s=a_tau_1+a_tau_2+...+a_tau_k-1 total shift sequence for any binary sequence a. The k-autocorrelation measures the similarity between the sequence a and the total shift sequence s. We give two application of the k-autocorrelation. In the first application, we would like to motivate our definition by providing an example related to design theory. In this specific example, we will explain the relation between k-autocorrelation coefficients of a binary sequence and corresponding lines in Fano plane. The second application is the circulant additive codes over F_4 in coding theory. We use the k-autocorrelation to determine the minimum distance of circulant additive codes over F_4.