Self-dual additive codes

Abstract

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group ${\mathbb {Z}}_{p^e}$. They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with $p \equiv 1 \pmod {4}$ and e is odd with $e>1$; and all lengths that are $0 \pmod {4}$ when p is an odd prime with $p \equiv 3 \pmod {4}$ and e is odd with $e>1.$