Let m₁ and m₂ be two distinct positive integers with \(d=\gcd (m_{1}, m_{2})\). Let \(\mathbb {F}_{2^{m}}\) be the finite field with 2^m elements, where m = m₁m₂/d. In this paper, we investigate the exponential sums$$ S(a,b)=\sum\limits_{x \in \mathbb{F}_{2^{m}}^{*}}(-1)^{\text{Tr}_{m_{1}}(ax^{\frac {2^{m}-1}{2^{m_{1}}-1}})+\text{Tr}_{m_{2}}(bx^{\frac {2^{m}-1}{2^{m_{2}}-1}})}, $$where \(a \in \mathbb {F}_{2^{m_{1}}}\), \(b \in \mathbb {F}_{2^{m_{2}}}\), and Tr_t denotes the trace function from \(\mathbb {F}_{2^{t}}\) to \(\mathbb {F}_{2}\). 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.

中文翻译：

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一类指数和和序列族

令m ₁和m ₂是两个具有\（d = \ gcd（m_ {1}，m_ {2}）\）的不同正整数。令\（\ mathbb {F} _ {2 ^ {m}} \）是具有2 ^m个元素的有限域，其中m = m ₁ m ₂ / d。在本文中，我们研究指数总和$$ S（a，b）= \ sum \ limits_ {x \ in \ mathbb {F} _ {{2 ^ {m}} ^ {*}}（-1）^ { \ text {Tr} _ {m_ {1}}（ax ^ {\ frac {2 ^ {m} -1} {2 ^ {m_ {1}}-1}}）+ \ text {Tr} _ {m_ {2}}（bx ^ {\ frac {2 ^ {m} -1} {2 ^ {m_ {2}}-1}}）}}，$$其中\（a \ in \ mathbb {F} _ { 2 ^ {m_ {1}}} \），\（b \ in \ mathbb {F} _ {2 ^ {m_ {2}}} \）和Tr _t表示从\（\ mathbb {F} _ {2 ^ {t}} \）到\（\ mathbb {F} _ {2} \）的跟踪函数。当d = 1，2，3，4时，我们明确给出指数和S（a，b）的值分布。作为应用程序，我们构造了具有三值相关性的三个二进制序列族。