A useful tool for constructing linear codes (Q2042936)
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scientific article; zbMATH DE number 7373702
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A useful tool for constructing linear codes |
scientific article; zbMATH DE number 7373702 |
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A useful tool for constructing linear codes (English)
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22 July 2021
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The authors introduce and discuss an elementary tool from the representation theory of finite groups that can be used to construct linear codes invariant under a given permutation group. The tool can be used to achieve theoretical insight as well as a way to explicitly determine generator matrices and weight distributions of codes. The authors use this tool to find, study and sometimes classify several types of codes invariant under various permutation groups, including new codes and codes found previously using graph-theoretical methods. The former include codes obtained by considering invariances under the actions of the Mathieu group \(M_{24}\), the Conway simple group \(\textrm{Co}_1\), and some of their subgroups. The latter include binary codes related to triangular graphs, and a 23-dimensional code constructed by Hans-Jörg Schaeffer in 1980 but not previously published.
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linear code
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Hamming weight
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weight distribution
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dual code
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MacWilliams' identities
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automorphism group
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permutation group
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symmetric permutation group
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Mathieu groups
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Conway groups
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representation theory
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module
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dual module
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permutation module
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