Orthogonalities from Group Characters

We explore non-standard orthogonalities arising from the character table of a finite commutative group. These orthogonalities are used in algebraic coding theory to replace the standard Euclidean orthogonality and have a corresponding MacWilliams relation for them. We establish that for a finite commutative group G and any subgroups H and K of G with |H||K| = |G| that there exists an orthogonality with H = K. Additionally, we give families of orthogonalities that apply to any finite commutative group. We give numerous examples of orthogonalities for specific groups. Finally, we prove that the additive group of the finite field F22k with a symmetric duality M has a self-dual code of length 1