Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings

We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring $RG$ and as such must have an automorphism group that contains $G$ as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative $[72,36,16]$ Type~II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48.