No MacWilliams duality for codes over nonabelian groups
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Abstract
Dougherty, Kim, and Sol\'e [3] have asked whether there is a duality theory and a MacWilliams formula for codes over nonabelian groups, or more generally, whether there is any subclass of nonabelian groups which have such a duality theory. We answer this in the negative by showing that there does not exist a nonabelian group $G$ with a duality theory on the subgroups of $G^n$ for all $n$.
Keywords
Dual code
Subgroup lattice
MacWilliams identity
Iwasawa group
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References
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[1]J. Chifman, Note on direct products of certain classes of finite groups, Comm. Algebra 37(5) (2009) 1831–1842.
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[2]R. Dedekind, Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann. 48(4) (1897) 548–561.
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[3]S. T. Dougherty, J.-L. Kim, and P. Solé, Open problems in coding theory, Contemp. Math. 634 (2015) 79–99.
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[4]K. Iwasawa, Über die endlichen Gruppen und die Verbände ihrer Untergruppen, J. Fac. Sci. Imp. Univ. Tokyo Sect. I 4 (1941) 171–199.
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[5]R. Schmidt, Subgroup Lattices of Groups, De Gruyter Expositions in Mathematics, vol. 14, Walter de Gruyter, Berlin, 1994.
ARTICLE DETAILS
Received
06 April 2020
Published
15 January 2018
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Copyright (c) 2018 Michael Ryan Julian