Linear codes have wide applications in many fields such as data storage, communication, cryptography, combinatorics. As a subclass of linear codes, minimal linear codes can be used to construct secret sharing schemes with good access structures. In this paper, we first construct some new classes of linear codes by selecting definition set properly. Then, the lengths, dimensions and the weight distribution of the codes are determined by investigating whether the intersections of the supports of vectors and the definition sets are empty. Results show that both wide and narrow minimal linear codes are contained in the new codes. Finally, we extend some existing results to general cases.

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.

Boolean functions with a few Walsh spectral values have important applications in sequence ciphers and coding theory. In this paper, we first construct a class of Boolean functions with at most five-valued Walsh spectra by using the secondary construction of Boolean functions, in particular, plateaued functions are included. Then, we construct three classes of Boolean functions with five-valued Walsh spectra using Kasami functions and investigate the Walsh spectrum distributions of the new functions. Finally, three classes of minimal linear codes with five-weights are obtained, which can be used to design secret sharing scheme with good access structures.