A duality for nonabelian group codes

Abstract

In 1962, Jesse MacWilliams published a set of formulas for linear and abelian group codes that among other applications, were incredibly valuable in the study of self-dual codes. Now called the MacWilliams Identities, her results relate the weight enumerator and complete weight enumerator of a code to those of its dual code. A similar set of MacWilliams identities has been proven to exist for many other types of codes. In 2013, Dougherty, Sol\'{e}, and Kim published a list of fundamental open questions in coding theory. Among them, Open Question 4.3: "Is there a duality and MacWilliams formula for codes over non-Abelian groups?" In this paper, we propose a duality for nonabelian group codes in terms of the irreducible representations of the group. We show that there is a Greene's Theorem and MacWilliams Identities which hold for this notion of duality. When the group is abelian, our results are equivalent to existing formulas in the literature.