Several classes of linear codes with few weights derived from Weil sums

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${\mathbb{F}}_q$ are presented with the defining sets given by the intersection, difference and union of two certain sets, where $t=3,4,5,6$ and $q$ is an odd prime power. By using Weil sums and Gauss sums, the parameters and weight distributions of these codes are determined completely. Moreover, three classes of optimal codes meeting the Griesmer bound are obtained, and computer experiments show that many (almost) optimal codes can be derived from our constructions.