Linear codes are widely studied due to their important applications in secret sharing schemes, authentication codes, association schemes and strongly regular graphs, etc. In this paper, firstly, a class of three-weight linear codes is constructed by selecting a new defining set, whose weight distributions are determined by exponential sums. Results show that almost all the constructed codes are minimal and thus can be used to construct secret sharing schemes with sound access structures. Particularly, a class of projective two-weight linear codes is obtained and based on which a strongly regular graph with new parameters is designed.

The calculation of cross-correlation between a sequence with good autocorrelation and its decimated sequence is an interesting problem in the field of sequence design. In this letter, we consider a class of ternary sequences with perfect autocorrelation, proposed by Shedd and Sarwate (IEEE Trans. Inf. Theory, 1979, DOI: 10.1109/TIT.1979.1055998), which is generated based on the cross-correlation between m-sequence and its d-decimation sequence. We calculate the cross-correlation distribution between a certain pair of such ternary perfect sequences and show that the cross-correlation takes three different values.

In this paper, for an odd prime p such that p≡3 mod 4, odd n, and d=(pn+1)/(pk+1)+(pn-1)/2 with k|n, the value distribution of the exponential sum S(a,b) is calculated as a and b run through $mathbb{F}_{p^n}$. The sequence family $mathcal{G}$ in which each sequence has the period of N=pn-1 is also constructed. The family size of $mathcal{G}$ is pn and the correlation magnitude is roughly upper bounded by $(p^k+1)sqrt{N}/2$. The weight distribution of the relevant cyclic code C over $mathbb{F}_p$ with the length N and the dimension ${ m dim}_{mathbb{F}_p}mathcal{C}=2n$ is also derived.

For a prime p with p≡3 (mod 4) and an odd number m, the Bentness of the p-ary binomial function fa,b(x)=Tr1n(axpm-1)+Tr12 is characterized, where n=2m, a ∈ F*pn, and b ∈ F*p2. The necessary and sufficient conditions of fa,b(x) being Bent are established respectively by an exponential sum and two sequences related to a and b. For the special case of p=3, we further characterize the Bentness of the ternary function fa,b(x) by the Hamming weight of a sequence.

Binary maximal-length sequences (or m-sequences) are sequences of period 2m-1 generated by a linear recursion of degree m. Decimating an m-sequence {st} by an integer d relatively prime to 2m-1 leads to another m-sequence {sdt} of the same period. The crosscorrelation of m-sequences has many applications in communication systems and has been an important and well studied problem during more than 40 years. This paper presents an updated survey on the crosscorrelation between binary m-sequences with at most five-valued crosscorrelation and shows some of the many recent connections of this problem to several areas of mathematics such as exponential sums and Dickson polynomials.

An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring ZN , N=pq for two distinct odd primes p and q. The result is a generalization of an estimate of exponential sums on rational point groups of elliptic curves over finite fields. The bound is applied to showing the pseudorandomness of a large family of binary sequences constructed by using elliptic curves over ZN .