Several classes of linear codes with few weights over finite fields

Abstract

Linear codes with few weights have applications in secret sharing schemes, authentication codes, association schemes and strongly regular graphs. In this paper, we introduce a new flexible defining set, and consequently, we obtain several classes of t-weight linear codes over Fp, where t=4,5,6 and p is an odd prime. The parameters and weight distributions of the generated linear codes are fully determined using quadratic Gauss sums over finite fields. Moreover, two classes of optimal two-weight codes meeting the Griesmer bound are obtained from our constructions. Our results are heavily based on evaluating some exponential sums over finite fields. Computer experiments using MAGMA programs show that many (almost) optimal codes can be derived from our results.