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
and any
subgroups
and
of
with
, there exists an
orthogonality that gives
.
The second uses a counting argument to show that the additive group of the finite field
with any
duality
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.
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