Projector additive group codes

Abstract

Let $F=\mathbb{F}_q$ and let $K=\mathbb{F}_{q^m}$ be a finite extension. An additive left group code is a left $FG$-submodule of the group algebra $KG$. In this paper, we introduce projector additive left group codes and restricted projector additive left group codes as additive counterparts of idempotent group codes in the classical theory of group codes. More precisely, they are defined, respectively, as images of $FG$-linear projectors on $KG$ and as images of left $FG$-submodules under such projectors. This perspective is motivated by the fact that idempotent elements of $KG$ do not yield a sufficiently general and natural algebraic framework for the study of additive left group codes. Projector additive left group codes are a particular class of projective left $FG$-submodules of $KG$. Consequently, in the semisimple case every additive left group code arises in this way, whereas in the non-semisimple case the projector construction captures precisely the direct summands of $KG$ as left $FG$-modules, and hence a natural subclass of projective left $FG$-submodules. We further relate trace-Euclidean and trace-Hermitian duality to adjoint projectors, establish criteria for the LCD and self-dual cases, study the Murray--von Neumann equivalence of projectors, and interpret quotients by orthogonal codes in terms of module duals.