Projector additive group codes

TL;DR

This paper introduces projector additive group codes as a natural extension of classical group codes, providing a framework for their algebraic structure and duality properties, especially in non-semisimple cases.

Contribution

It defines projector additive left group codes and relates them to module theory, duality, and direct summands, broadening the algebraic understanding of additive group codes.

Findings

In the semisimple case, all additive left group codes are captured by projector codes.

The paper establishes criteria for LCD and self-dual codes within this framework.

It explores the Murray--von Neumann equivalence of projectors and quotient interpretations.

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.