Linear codes with a few weights have many applications in secret sharing schemes, authentication codes, association schemes and strongly regular graphs, and they are also of importance in consumer electronics, communications and data storage systems. In this paper, based on the theory of defining sets, we present a class of five-weight linear codes over $mathbb{F}_p$(p is an odd prime), which include an almost optimal code with respect to the Griesmer bound. Then, we use exponential sums to determine the weight distribution.

A family of quaternary sequences over Z4 is defined based on the Ding-Helleseth generalized cyclotomic classes modulo pq for two distinct odd primes p and q. The linear complexity is determined by computing the defining polynomial of the sequences, which is in fact connected with the discrete Fourier transform of the sequences. The results show that the sequences possess large linear complexity and are “good” sequences from the viewpoint of cryptography.