Weight distributions and weight hierarchies of a class of binary linear codes with a few weights

Linear codes can be applied in consumer electronic, communica-tion and data storage system. In addition, the weight distribution and weight hierarchy of a linear code are two significant research topics in coding the-ory. In this paper, by extending Li et al.'s construction in [Finite Fields Appl. 73(2021), 101865], a class of binary linear codes with a few weights is con-structed by selecting the defining set properly, then the weight distributions and weight hierarchies are determined by calculating the corresponding expo-nential sums. In particular, the codes constructed in this paper are m-MDS codes and minimal and thus can be used in secret sharing schemes.