在编码和密码学的研究中涉及到很多有限域上的特征和（指数和）的问题。例如一些循环码的参数的确定本质上可以归结到某些特征和的计算；一些编码体制的安全性等很多问题归结到一些代数方程（组）的求解问题，最终也可以归结到一些特征和的计算。有限域上的特征和研究在有限域的理论中占中心地位，同时也是代数数论和算术几何中重要的研究对象和工具。虽然人们知道一些特征和的值或者上下界和均值，但是能够给出显式表示的不多见。本课题研究在编码密码中涉及到的几种有限域上的特征和问题。试图给出一些指数和的显式表示或者值分布。对一些特殊的指数和，给出某个具体值出现的次数。利用这些结果，我们可以给出一些线性码的权分布及高维权分布等问题，也可以构造出一些最优线性码或者良相关序列，还可以构造一些（广义）Bent函数和差集，强正则图等。

There are many exponential sums involved in the studying on cryptography and coding theory. For example, the determination of the weight distribution of some cyclic codes, the security of some protocols,and the solutions of some equations over finite fields, are all run into the solution of a question: the evaluation of some exponential sums. Thus, the reaserching of exponential sums stands on the central stage of the studying of finite fields, and is also a main tool and target in algebraic number theory and arithmatic geometry. However, we only know some lower or upper bounds of some exponential sums,except for a few of explicit expressions of some characterisitc sums. The main purpose of this project is investigating some exponential sums involved in coding and cryptography,we will try to give the explicit expression or value distribution of some exponential sums. For some specific exponential sums, we will give the frequency of certain values.Using these results, we can obtian the weight distribution and high dimentional weight distributions of some linear codes, and construct some optimal linear codes and sequences with ideal correlations. We can also construct some (generalized) bent functions and difference sets, strongly regular graphs,etc.