Vector Network Coding Based on Subspace Codes Outperforms Scalar Linear Network Coding

TL;DR

This paper demonstrates that vector network coding based on subspace and rank-metric codes can significantly reduce the required field size compared to scalar linear network coding in multicast networks, especially as network parameters grow.

Contribution

It introduces a new theoretical result showing a large gap in field size between vector and scalar network coding, extending previous constant-gap findings.

Findings

Vector network coding reduces field size compared to scalar coding

The gap in field size grows exponentially with network parameters

Results apply to various multicast network configurations

Abstract

This paper considers vector network coding based on rank-metric codes and subspace codes. Our main result is that vector network coding can significantly reduce the required field size compared to scalar linear network coding in the same multicast network. The achieved gap between the field size of scalar and vector network coding is in the order of $q^{(\ell-1)t^2/\ell}-q^t$, for any $q \geq 2$, where $t$ denotes the dimension of the vector solution, and the number of messages is $2 \ell$, ${\ell \geq 2}$. Previously, only a gap of constant size had been shown. This implies also the same gap between the field size in linear and non-linear scalar network coding for multicast networks. Similar results are given for any number of odd messages greater than two. The results are obtained by considering several multicast networks which are variations of the well-known combination network.