A code \(C\) is formally self-dual if \(C\) has the same weight distribution as its dual code \(C^ \perp\). The authors study binary formally self- dual codes and demonstrate that the class of such codes contains codes that have greater minimum distance than any self-dual code with the same parameters. A strengthening of the Assmus-Mattson theorem that yields designs formed by words in \(C\cup C^ \perp\) is proved.