Linear complexity and correlation of a class of binary cyclotomic sequences
Let \(p_1,p_2,\ldots ,p_n\) be distinct odd primes and let \(e_1,e_2,\ldots ,e_n\) be positive integers. Based on cyclotomic classes proposed by Ding and Helleseth (Finite Fields Appl 4:140166, 1998), a binary cyclotomic sequence of period \(p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}\) is defined and denoted by \(\underline{s_\Upsilon }\). The linear complexity of \(\underline{s_\Upsilon }\) is determined and is proved to be greater than or equal to \((p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}-1)/2\). The autocorrelation function of \(\underline{s_\Upsilon }\) is explicitly computed. Let \(\ell \in \{1,2,\ldots ,n\}\). We also explicitly compute the crosscorrelation function of \(\underline{s_\Upsilon }\) and the Legendre sequence \(\underline{L_{p_\ell }}\) with respect to \(p_\ell \). It is shown that \(\underline{s_\Upsilon }\) and \(\underline{L_{p_\ell }}\) have two-level or three-level crosscorrelation, and all their two-level crosscorrelation functions are determined.
linear complexity cyclotomic sequence autocorrelation crosscorrelation legendre symbol 11b50 11t22 11t71