Exponential sums, Gauss sums and cyclic codes (Q2713279)

Let \(F\) be a finite field and let \(e\) be the canonical additive character of \(F\). An exponential sum over \(F\) is of the form \(S(f)= \sum_{x\in F} e(f(x))\) with \(f(x)\in F[x]\). This dissertation considers \(S(f)\) in the cases when \(f(x)\) is a monomial of the form \(f(x)= \alpha x^N\) or a binomial of the form \(f(x)= \alpha x^N+ \beta x\). Several applications of these results to the weight distribution of cyclic codes are also given. NEWLINENEWLINENEWLINEThe dissertation is divided into three papers preceded by an introduction giving an overview of the main results. In the first paper relations between multiple Kloosterman sums and certain monomial and binomial exponential sums are presented. In the second paper exponential sums and Gauss sums are calculated in the case when \(f(x)= \alpha x^N\) and \(N=p^\alpha q^\beta\) when the order of the characteristic of \(F\) modulo \(N\) is \(\varphi(N)/2\) (the index two case). The third paper presents two recursive algorithms for computing the weight distribution of certain binary irreducible cyclic codes of length \(n= (2^m-1)/N\) and dimension \(k= \varphi(N)/2\) corresponding to the index two case. The algorithm is illustrated by presenting tables for codes of dimensions less than 200.