#### Vol. 14, No. 4, 2021

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Orthogonality from group characters

### Steven T. Dougherty and Sara Myers

Vol. 14 (2021), No. 4, 555–570
##### 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 ${H}^{\perp }=K$. The second uses a counting argument to show that the additive group of the finite field ${\mathbb{𝔽}}_{{2}^{2k}}$ 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