Codes over a family of local Frobenius rings, Gray maps and self-dual codes

Abstract

We define a class of finite Frobenius rings of order 22k, describe their generating characters, and study codes over these rings. We define two conjugate weight preserving Gray maps to the binary space and study the images of linear codes under these maps. This structure couches existing Gray maps, which are a foundational idea in codes over rings, in a unified structure and produces new infinite classes of rings with a Gray map. The existence of self-dual and formally self-dual codes is determined and the binary images of these codes are studied.