On a duality for codes over non-abelian groups

Heiko Dietrich, Jeroen Schillewaert

Research output: Contribution to journalArticleResearchpeer-review

Abstract

This work is motivated by a well-known open problem in coding theory, asking whether there is a duality theory for codes over non-abelian groups, see Dougherty et al. (Contemp Math 634:79–99, 2015). We prove that such a duality cannot be induced by a duality of a group lattice, and then study a variation that reduces to a group theoretic investigation: We say a finite group of order m has a layer-symmetric lattice if for every divisor d of m there is a bijection between the subgroups of order d and the subgroups of order m/d. We prove that every such group is nilpotent, and then investigate the class of finite p-groups with a layer-symmetric lattice.
Original languageEnglish
Number of pages17
JournalDesigns Codes and Cryptography
DOIs
Publication statusAccepted/In press - 2020

Keywords

  • Group codes
  • Duality
  • Group lattice
  • Nilpotent groups

Cite this

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On a duality for codes over non-abelian groups. / Dietrich, Heiko; Schillewaert, Jeroen.

In: Designs Codes and Cryptography, 2020.

Research output: Contribution to journalArticleResearchpeer-review

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T1 - On a duality for codes over non-abelian groups

AU - Dietrich, Heiko

AU - Schillewaert, Jeroen

PY - 2020

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N2 - This work is motivated by a well-known open problem in coding theory, asking whether there is a duality theory for codes over non-abelian groups, see Dougherty et al. (Contemp Math 634:79–99, 2015). We prove that such a duality cannot be induced by a duality of a group lattice, and then study a variation that reduces to a group theoretic investigation: We say a finite group of order m has a layer-symmetric lattice if for every divisor d of m there is a bijection between the subgroups of order d and the subgroups of order m/d. We prove that every such group is nilpotent, and then investigate the class of finite p-groups with a layer-symmetric lattice.

AB - This work is motivated by a well-known open problem in coding theory, asking whether there is a duality theory for codes over non-abelian groups, see Dougherty et al. (Contemp Math 634:79–99, 2015). We prove that such a duality cannot be induced by a duality of a group lattice, and then study a variation that reduces to a group theoretic investigation: We say a finite group of order m has a layer-symmetric lattice if for every divisor d of m there is a bijection between the subgroups of order d and the subgroups of order m/d. We prove that every such group is nilpotent, and then investigate the class of finite p-groups with a layer-symmetric lattice.

KW - Group codes

KW - Duality

KW - Group lattice

KW - Nilpotent groups

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DO - 10.1007/s10623-019-00711-z

M3 - Article

JO - Designs Codes and Cryptography

JF - Designs Codes and Cryptography

SN - 0925-1022

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