The weight distribution of a class of <i>p</i>-ary cyclic codes and their applications

Cyclic codes over finite field have been studied for decades due to their wide applications in communication and storage systems. However their weight distributions are known only in a few cases. In this paper, we investigate a class of $ p$-ary cyclic codes whose duals have three zeros, where $ p$ is an odd prime. The weight distributions of the class of cyclic codes for all distinct cases are determined explicitly. The results indicate that these codes contain five-weight codes, seven-weight codes and eleven-weight codes. Some of these codes are optimal. Moreover, the covering structures of the class of codes are considered and being used to construct secret sharing schemes.