No MacWilliams duality for codes over nonabelian groups

Year 2018, Volume: 5 Issue: 1, 45 - 49, 15.01.2018

M. Ryan Julian Jr.

https://doi.org/10.13069/jacodesmath.369864

Cited By: 3

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

References

• [1] J. Chifman, Note on direct products of certain classes of finite groups, Commun. Algebra 37(5) (2009) 1831–1842.
• [2] R. Dedekind, Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann. 48(4) (1897) 548–561.
• [3] S. Dougherty, J.-L. Kim, P. Solé, Open problems in coding theory, Contemp. Math. 634 (2015) 79–99.
• [4] K. Iwasawa, Über die endlichen Gruppen und die Verbände ihrer Untergruppen, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4 (1941) 171–199.
• [5] R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter, Berlin, 1994.
• [6] M. Suzuki, On the lattice of subgroups of finite groups, Trans. Amer. Math. Soc. 70(2) (1951) 345–371.
• [7] G. Zacher, Caratterizzazione dei gruppi immagini omomorfe duali di un gruppo finito, Rend. Sem. Mat. Univ. Padova 31 (1961) 412–422.