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In this work, we determine new linear equations for the weight distribution of linear codes over finite chain rings. The identities are determined by counting the number of some special submatrices of the parity-check matrix of the code. Thanks to these relations we are able to compute the full weight distribution of codes with small Singleton defects, such as MDS, MDR and AMDR codes. [ABSTRACT FROM AUTHOR]

In this paper we present the Multidimensional Byzantine Agreement (MBA) Protocol, a leaderless Byzantine agreement protocol defined for complete and synchronous networks that allows a network of nodes to reach consensus on a vector of relevant information regarding a set of observed events. The consensus process is carried out in parallel on each component, and the output is a vector whose components are either values with wide agreement in the network (even if no individual node agrees on every value) or a special value ⊥ that signals irreconcilable disagreement. The MBA Protocol is probabilistic and its execution halts with probability 1, and the number of steps necessary to halt follows a Bernoulli-like distribution. The design combines a Multidimensional Graded Consensus and a Multidimensional Binary Byzantine Agreement, the generalization to the multidimensional case of two protocols presented by Micali et al. (SIAM J Comput 26(4):873–933, 1997; Byzantine agreement, made trivial, 2016). We prove the correctness and security of the protocol assuming a synchronous network where less than a third of the nodes are malicious. [ABSTRACT FROM AUTHOR]

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations. [ABSTRACT FROM AUTHOR]

Blockchain-based platforms utilise cryptographic protocols to enforce the correct behaviour of users, as well as to guarantee a sufficient level of protection against malicious adversaries. Cryptography is, however, an ever-evolving discipline, and any breakthrough would have immediate consequences on the security of blockchain-based applications. A possible threat currently under investigation is given by the development of quantum computers, since several wide-adopted cryptographic protocols have been proved to be unsafe against quantum-capable adversaries. In this work, we propose a novel approach for the management of cryptographic primitives in smart-contract-based ledgers, discussing how it fits in both a (partially) permissioned and a fully permissionless setting. The cryptographic protocols are managed in a flexible manner via a set of smart-contracts defined on the ledger itself, in this way the choice of algorithms and parameters can change quickly. Among the advantages of this approach, we remark how it allows designing an adaptive post-quantum-based blockchain that keeps up with ongoing technological advances. In general, the introduction of new features and the application of fixes to a blockchain cause forks in the chain, which may cause major disruptions. The use of smart contracts in blockchain management allows to avoid this problem, dynamically introducing new protocols or deprecating old ones without compromising previous data. The Cryptographic Kernel approach has been adopted by Quadrans, an open-source, public, decentralised smart-contract-based blockchain with a specific focus on the needs of industry, complex supply chains, and IOT devices. [ABSTRACT FROM AUTHOR]