Minimal linear codes have significant applications in secret sharing schemes and secure twoparty computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many construction methods for linear codes from functions have been proposed in the literature. In this paper, we generalize the recent construction methods given by Tang et al. in IEEE Transactions on Information Theory, 62(3), 1166-1176, 2016 to weakly regular plateaued functions over finite fields of odd characteristic. We first construct threeweight linear codes from weakly regular plateaued functions based on the second generic construction and then determine their weight distributions. We also give a punctured version and subcode of each constructed code. We note that they may be (almost) optimal codes and can be directly employed to obtain (democratic) secret sharing schemes, which have diverse applications in the industry. We next observe that the constructed codes are minimal for almost all cases and finally describe the access structures of the secret sharing schemes based on their dual codes.

Blockchain systems store transaction data in the form of a distributed ledger where each node stores a copy of all data, which gives rise to storage issues. It is well-known that the tremendous storage and distribution of the block data are common problems in blockchain systems. In the literature, some types of secret sharing schemes are employed to overcome these problems. The secret sharing method is one of the most significant cryptographic protocols used to ensure the privacy of the data. The main purpose of this paper is to improve the recent distributed storage blockchain systems by proposing an alternative secret sharing method. We first propose a secure threshold verifiable multi-secret sharing scheme that has the verification and private communication steps based on post-quantum lattice-based hard problems. We then apply the proposed threshold scheme to the distributed storage blockchain (DSB) system to share transaction data at each block. In the proposed DSB system, we encrypt the data block with the AES-256 encryption algorithm before distributing it among nodes at each block, and both its secret key and the hash value of the block are privately shared among nodes simultaneously by the proposed scheme. Thereafter, in the DSB system, the encrypted data block is encoded by the Reed–Solomon code, and it is shared among nodes. We finally analyze the storage and recovery communication costs and the robustness of the proposed DSB system. We observe that our approach improves effectively the recovery communication cost and makes it more robust compared to the previous DSB systems. It also improves extremely the storage cost of the traditional blockchain systems. Furthermore, the proposed scheme brings to the DSB system the desirable properties such as verification process and secret communication without private channels in addition to the known properties of the schemes used in the previous DSB systems. As a result of the flexibility on the threshold parameter of the scheme, a diverse range of qualified subsets of nodes in the DSB system can privately recover the secret values.