Self-dual codes with group actions and invariants

In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden’s theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the concept of $G$-dual codes for codes invariant under a permutation group $G$, referred to as $G$-codes. We then present several generalizations of Atsumi’s MacWilliams identity (Atsumi, 1995; Chakraborty and Miezaki, 2023) for $G$-codes over finite rings with respect to general bilinear forms. Furthermore, we establish a $G$-analogue of the MacWilliams identity for $G$-full weight enumerators and introduce the notions of $G$-quadratic maps and $G$-representations for twisted modules, twisted rings, quadratic pairs, and form rings. By defining transformation groups for $G$-full weight enumerators, we extend the theory of Clifford–Weil groups (Nebe, Rains, and Sloane, 2004, 2006). Finally, we provide generalizations of Gleason-type theorems for these weight enumerators, demonstrating that the $G$-full weight enumerators of $G$-self-dual and $G$-isotropic codes are invariant under the Clifford–Weil groups and span the invariant subspaces of these groups.