On Base Field of Linear Network Coding

TL;DR

This paper introduces a new class of multicast networks and provides an explicit formula linking their linear solvability to algebraic properties of the base field, revealing infinitely many networks solvable over certain fields but not others.

Contribution

The paper designs a new class of multicast networks with a formula connecting solvability to subgroup coset numbers, unveiling infinitely many networks with field-dependent solvability.

Findings

Explicit formula for linear solvability involving coset numbers

Existence of infinitely many networks solvable over GF(q) but not GF(q')

Construction of networks with Θ(q^2) nodes and edges for minimal field size

Abstract

For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class $\mathcal{N}$ of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil \emph{infinitely many} new multicast networks linearly solvable over GF($q$) but not over GF($q'$) with $q < q'$, based on a subgroup order criterion. In particular, i) for every $k\geq 2$, an instance in $\mathcal{N}$ can be found linearly solvable over GF($2^{2k}$) but \emph{not} over GF($2^{2k+1}$), and ii) for arbitrary distinct primes $p$ and $p'$, there are infinitely many $k$ and $k'$ such that an instance in $\mathcal{N}$ can be found linearly solvable over GF($p^k$) but \emph{not} over GF($p'^{k'}$) with $p^k < p'^{k'}$. On the other hand, the construction of $\mathcal{N}$ also leads to a new class of multicast networks with $\Theta(q^2)$ nodes and $\Theta(q^2)$ edges, where $q \geq 5$ is the minimum field size for linear solvability of the network.