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

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Linear codes can be applied in consumer electronic, communication and data storage system. In addition, the weight distribution and weight hierarchy of a linear code are two significant research topics in coding theory. 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 constructed by selecting the defining set properly, then the weight distributions and weight hierarchies are determined by calculating the corresponding exponential sums. In particular, the codes constructed in this paper are m-MDS codes and minimal and thus can be used in secret sharing schemes.