Orthogonality from group characters

Dougherty, Steven T.; Myers, Sara

doi:10.2140/involve.2021.14.555

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Orthogonality from group characters

Steven T. Dougherty and Sara Myers

Vol. 14 (2021), No. 4, 555–570

DOI: 10.2140/involve.2021.14.555

Abstract

We prove two major results about using group characters to define orthogonality for codes over abelian groups. The first is that for a finite commutative group $G$ and any subgroups $H$ and $K$ of $G$ with $| H | | K | = | G |$ , there exists an orthogonality that gives $H^{⊥} = K$ . The second uses a counting argument to show that the additive group of the finite field $𝔽_{2^{2 k}}$ with any duality $M$ has a self-dual code of length 1 and therefore of all lengths. Additionally, we give numerous examples of orthogonalities for specific groups and we give families of orthogonalities that apply to any finite commutative group.

Keywords

self-dual codes, group characters, orthogonality

Mathematical Subject Classification 2010

Primary: 11T71, 94B05

Milestones

Received: 31 May 2018

Revised: 6 February 2021

Accepted: 31 March 2021

Published: 23 October 2021

Communicated by Kenneth S. Berenhaut

Authors

Steven T. Dougherty
Department of Mathematics University of Scranton Scranton, PA United States

Sara Myers
Department of Mathematics University of Scranton Scranton, PA United States

Vol. 14, No. 4, 2021