Constructions Of Bent Functions, Weight Distributions Of Cyclic Codes And Related Questions

With the high-speed development of computer and Internet technology, cryptography and coding theory are playing a more and more important role in practical application, which attracts many researchers especially in the fields of constructions of nonlinear functions, weight distribu-tions of cyclic codes and the structure of additive codes etc. Based on the previous work, we study several classes of (generalized) polynomial functions with Dillom index and determine the weight distributions of two classes of cyclic codes. We also discuss the structure and properties of one weight Z2Z4-additive codes. The main results are given as follows:In Chapter 2, we construct several classes of (generalized) polynomial functions with Dillon index. By using partition of cyclotomic cosets, we study some partial exponential sums. The bentness of these (generalized) polynomial functions is characterized by some partial exponential sums, which have close relations with Kloosterman sums. Furthermore, we obtain some new (generalized) Bent functions. In particular, by taking suitable values of parameters, the bentness of some kinds of (generalized) polynomial functions is determined by Kloosterman sums.In Chapter 3, by using quadratic form, we determine the weight distributions of the following two classes of cyclic codes Ct and C over the finite field Fp, where p is prime. Let m be a positive integer, tt be the primitive element of the finite field m.(i) Let t satisfy t≡(pk+1)/2-pT (mod (pm-1)/2), where k is a positive integer, τ∈Zm. Let h1(x) and h2(x) be the minimal polynomials of π-1 and -π-t over Fp, respectively. The cyclic code with parity-check polynomial h1(x)h2(x) can be expressed as(ii) Let 1≤v2(m)< v2(k) or v2(k)< v2(m), where m, k are both positive integers. Let v2(j) denote the 2-adic valuation of j and let h1(x) and h2(x) be the minimal polynomials of π-(pk+1)/2 and-π-1 over Fp, respectively. The cyclic code with parity-check polynomial hi(x)h2(x) is defined to beIn Chapter 4, we study one weight Z2Z4-additive codes. The structure of one weight Z2Z4-additive codes is described. We obtain the relation between possible weights for all one weight Z2Z4-additive codes and parameters. A lower bound for the minimum distance of dual codes of one weight additive codes is obtained. In addition, a necessary and sufficient condition on the image of one weight Z2Z4-additive codes to be linear under the Gray map is also obtained.