In this paper, we employ group rings and skew group rings to construct binary self-dual codes of length 72. There is a well known map \(\sigma \) that sends a (skew) group ring element to an \(n\times n\) matrix. By using groups of order n, we obtain a lot of generator matrices of the form \((I_{n}\mid \sigma _{\varphi }(v)\) ), where v is an element in a (skew) group ring. Then through special maps, we can send a self-dual code of length 2n over \({\mathbb {F}}_{2^t}\) to a binary self-dual code of length 2tn. We use these generator matrices to search for binary [72,36,12] self-dual codes and obtain many singly-even and doubly-even codes with new parameters in their weight enumerators that were not known in the literature before. We list our findings on a publicly available website.
