No MacWilliams duality for codes over nonabelian groups
Keywords:
Dual code, Subgroup lattice, MacWilliams identity, Iwasawa group
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 $n$.
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R. Dedekind, Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann. 48(4) (1897) 548–561.
S. Dougherty, J.-L. Kim, P. Solé, Open problems in coding theory, Contemp. Math. 634 (2015) 79–99.
K. Iwasawa, Über die endlichen Gruppen und die Verbände ihrer Untergruppen, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4 (1941) 171–199.
R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter, Berlin, 1994.
M. Suzuki, On the lattice of subgroups of finite groups, Trans. Amer. Math. Soc. 70(2) (1951) 345–371.
G. Zacher, Caratterizzazione dei gruppi immagini omomorfe duali di un gruppo finito, Rend. Sem. Mat. Univ. Padova 31 (1961) 412–422.
