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 [11] 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.

In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings R k and the ring F 4 + uF 4 . We use groups of order 4, 12 and 20. We construct some extremal self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68.

We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + uF 2 , using groups of * corresponding author orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.