Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

A class of binary cyclic codes with generalized Niho exponents

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined.     

The weight distributions of two classes of p-ary cyclic codes with few weights

Cyclic codes have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms. In this paper, for an odd prime p, we investigate two classes of p-ary cyclic codes for special cases and determine their weight distributions explicitly. The results show that both codes have at most five nonzero weights.     

Mean value of mixed exponential sums

For integers , , , with , and Dirichlet character , we define a mixed exponential sum

where , and denotes the summation over all with . The main purpose of this paper is to study the mean value of

and to give a related identity on the mean value of the general Kloosterman sum

where .

     

Estimate for exponential sums and its applications

In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-Hooley contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established.     

Irreducibility and the distribution of some exponential sums

We relate the distribution of the absolute value of some generalized Gauss sums to the absolute irreducibility of some polynomials in two variables in characteristic 0 and p.     

Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k be integers with q?3 and k?1, define the exponential sum

     

On some congruence with application to exponential sums

We will study the solution of a congruence,x ≡g ^1/2)ωg(2 ⁿ ) mod 2 ⁿ , depending on the integersg andn, where ω _g (2 ⁿ ) denotes the order ofg modulo 2 ⁿ . Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.     

Divisibility properties of classical binary Kloosterman sums

Let K(a) be the so-called classical Kloosterman sum over . In this paper, we compute K(a) modulo 24 for even m, completing our previous results for odd m. We extensively study the links between K(a) and other exponential sums, especially the cubic sums. We point out (as we did for odd m) that the values K(a) are involved in the computation of the weight distributions of cosets of primitive narrow sense extended BCH codes of length 2^m and minimum distance 8. We also complete some recent results on K(a)−1 modulo 3.     

Linear codes with few weights from weakly regular plateaued functions

Linear codes with few nonzero weights have wide applications in secret sharing, authentication codes, association schemes and strongly regular graphs. Recently, Wu et al. (2020) obtained some few-weighted linear codes by employing bent functions. In this paper, inspired by Wu et al. and some pioneers' ideas, we use a kind of functions, namely, general weakly regular plateaued functions, to define the defining sets of linear codes. Then, by utilizing some cyclotomic techniques, we construct some linear codes with few weights and obtain their weight distributions. Notably, some of the obtained codes are almost optimal with respect to the Griesmer bound. Finally, we observe that our newly constructed codes are minimal for almost all cases.     

Approximation of exponential sums by shorter ones

A new theorem on approximation of exponential sum by shorter one is proved.