Permute & Add Network Codes via Group Algebras

TL;DR

This paper introduces a novel group algebra framework for designing permute-and-add network codes applicable to arbitrary networks, enabling flexible trade-offs between coding rate and complexity.

Contribution

It extends permute-and-add network codes beyond multicast networks using group algebra, allowing arbitrary permutation groups and trade-offs between rate and degree.

Findings

Framework generalizes existing permute-and-add codes

Supports arbitrary permutation groups including circular shifts

Enables rate-degree trade-offs in code design

Abstract

A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as "permute-and-add" network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their "degree" which is the number of permutations applied on each incoming vector to compute an outgoing vector. Constructions of permute-and-add network codes for multicast networks are known. In this paper, we provide a new framework based on group algebras to design permute-and-add network codes for arbitrary (not necessarily multicast) networks. Our framework allows the use of any finite group of permutations (including circular shifts, proposed in prior works) and admits a trade-off between coding rate and the degree of the code. Further, our technique permits elegant recovery and generalizations of the key results on permute-and-add network codes known in the literature.