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.