A class of exponential sums and sequence families

Let m 1 and m 2 be two distinct positive integers with d= (m_1, m_2) . Let 𝔽_2^m be the finite field with 2 m elements, where m = m 1 m 2 / d . In this paper, we investigate the exponential sums S(a,b)=∑_x ∈𝔽_2^m^*(-1)^Tr_m_1(ax^2^m-1/2^m_1-1)+Tr_m_2(bx^2^m-1/2^m_2-1), where a ∈𝔽_2^m_1 , b ∈𝔽_2^m_2 , and Tr t denotes the trace function from 𝔽_2^t to 𝔽_2 . When d = 1, 2, 3, 4, we present the value distribution of the exponential sums S ( a , b ) explicitly. As an application, we construct three families of binary sequences with three-valued correlation.