A class of binary sequences with two-valued cross correlations

Abstract

Let $ p $ be an odd prime with $ p\equiv 7\pmod{12} $, $ \frac{p-1}2 $ be the least integer such that $ 2^{\frac{p-1}2}\equiv 1\pmod p $, and $ q = 2^{\frac{p-1}2} $. Let $ \alpha $ be a primitive element of the finite field $ \Bbb F_{q} $ and $ \beta = \alpha^{\frac{q-1}{p}} $. Suppose that $ \sigma = \sum_{i = 0}^2\beta^{m\zeta_3^i}\in \Bbb F_q^* $, where $ m\in \Bbb F_p^* $ and $ \zeta_3 $ is a $ 3 $rd root of unity in $ \Bbb F_p $. Let $ \{u_i\} = (\operatorname{Tr}_{q/2}(\sigma\beta^i))_{i = 0}^{q-2} $ be a binary sequence of period $ q-1 $. In this paper, we obtained the cross correlation distribution between two sequences $ \{u_i\} $ and its $ \frac{q-1}p $-decimation sequence, which is two-valued.

Keywords

• cross correlation
• exponential sum