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New families of non-Reed-Solomon MDS codes
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abstract
MDS codes have garnered significant attention due to their wide applications in practice. To date, most known MDS codes are equivalent to Reed-Solomon codes. The construction of non-Reed-Solomon (non-RS) type MDS codes has emerged as an intriguing and important problem in both coding theory and finite geometry. Although some constructions of non-RS type MDS codes have been presented in the literature, the parameters of these MDS codes remain subject to strict constraints. In this paper, we introduce a general framework of constructing $[n,k]$ MDS codes using the idea of selecting a suitable set of evaluation polynomials and a set of evaluation points such that all nonzero polynomials have at most $k-1$ zeros in the evaluation set. Moreover, these MDS codes can be proved to be non-Reed-Solomon by computing their Schur squares. Furthermore, several explicit constructions of non-RS MDS codes are given by converting to combinatorial problems. As a result, new families of non-RS MDS codes with much more flexible lengths can be obtained and most of them are not covered by the known results.
Forward citations
Cited by 4 Pith papers
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On subcodes of the generalized Reed-Solomon codes
It characterizes self-duality and near-MDS status for one-codimensional subcodes of generalized Reed-Solomon codes for all r, and determines the dual codes for r=1,2,k-1.
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Row-Column Twisted Reed-Solomon codes
A new family of maximum-distance-separable codes, RCTRS, is built by applying row and column twists to Reed-Solomon codes and is claimed to be inequivalent to both RS and column-twisted RS codes.
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Shelby: Decentralized Storage Designed to Serve
Shelby is a decentralized storage protocol design that targets Web2-grade read performance using Clay codes, micropayments, a dedicated network backbone, and an 'audit-the-auditor' scheme.
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Some constructions of non-generalized Reed-Solomon MDS Codes
Necessary and sufficient conditions are given for two extended evaluation-code families to be non-GRS MDS codes, and o-monomials are characterized by nonvanishing complete symmetric functions.
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