Few-weight linear codes and their application to secret sharing

Abstract

In this paper, for an odd prime power q , we study a class of cyclic codes over $$\mathbb {F}_q$$ F q whose duals have two zeros. Using some results on Gaussian periods of order 2, along with some relations between certain exponential sums and multiple Kloosterman sums, we determine the weight distributions for these codes. We also present two classes of linear codes obtained by puncturing and shortening the aforementioned cyclic codes. Both the cyclic and linear codes have few weights, which is of interest since few-weight linear codes have applications in cryptography, particularly in secret sharing. Furthermore, we investigate the duals for these three classes of codes and find out that some of them are almost optimal with respect to the sphere-packing bound. Finally, we use some of the studied codes to construct secret-sharing schemes.