Orthogonality from group characters

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 𝔽22k 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.