Towards an Operational Definition of Group Network Codes

TL;DR

This paper establishes an operational framework for group network codes by connecting them to Coordinate-Wise-Linear codes, enabling a local encoding perspective for Abelian group codes.

Contribution

It introduces an operational definition for group network codes by linking them to CWL codes, bridging a gap in understanding local encoding functions.

Findings

Abelian group codes can be expressed as CWL codes

CWL codes generalize linear codes and are locally definable

Operational definitions for group codes are now possible

Abstract

Group network codes are a generalization of linear codes that have seen several studies over the last decade. When studying network codes, operations performed at internal network nodes called local encoding functions, are of significant interest. While local encoding functions of linear codes are well understood (and of operational significance), no similar operational definition exists for group network codes. To bridge this gap, we study the connections between group network codes and a family of codes called Coordinate-Wise-Linear (CWL) codes. CWL codes generalize linear codes and, in addition, can be defined locally (i.e., operationally). In this work, we study the connection between CWL codes and group codes from both a local and global encoding perspective. We show that Abelian group codes can be expressed as CWL codes and, as a result, they inherit an operational definition.