指数和在扩频序列设计中的应用

Applications of Exponential Sums to Spreading Sequences Designs

设ｐ为奇素数，ｎ、ｍ、ｅ、ｒ是满足如下条件的正整数：ｎ＝ｅｍ，１≤ｒ≤ｅ－１，（ｒ，ｅ）＝１，ω是ｐ次本原单位根，ｑ＝ｐｍ，Ψ：ｚ→ωｔｒｍ１（ｚ）是有限域Ｆｑ上加法特征．对任意ａ，ｂ∈Ｆｑｅ，定义指数和σ（ａ，ｂ）＝ｘ∈ＦｑｅΨ［ｔｒｎｍ（ａｘ＋ｂｘｑｒ＋１）］，其中ｔｒｎｍ（·）表示从Ｆｑｅ到Ｆｑ上的迹函数．文中求出了指数和σ（ａ，ｂ）的值，并讨论了该指数和在扩频序列设计中的应用．

Let p be an odd prime, n,e,m and r be integers satisfying the following conditions:n=em,1≤r≤e-1,(r,e)=1. Let ω be a complex primitive pth root of unity and Ψ:z→w tr m 1(z) be the nontrivial addition character of the finite field F q(q=p m). For every pair a,b∈F q e ,Let σ(a,b) denote the exponential sum: σ(a,b)=x∈F q e Ψ[tr n m(ax+bx q r+1 )], where tr n m(·) denotes the trace function of F q e over F q. In this paper we determined the values of the exponential sum σ(a,b) and discussed the applications of exponential sums in spreading sequences designs.