On Vector Linear Solvability of Multicast Networks

TL;DR

This paper systematically constructs multicast networks to demonstrate the advantages of vector linear network coding over scalar coding, confirming the conjecture for infinitely many alphabet sizes and revealing nuanced solution properties.

Contribution

It introduces a systematic construction method for multicast networks that affirm the conjecture and provides explicit examples with unique vector and scalar solution properties.

Findings

Vector linear solvability can differ from scalar solvability for certain networks.

Explicit networks exist that are vector solvable of dimension L but not scalar solvable.

Some networks have scalar solutions over smaller fields but no vector solutions of dimension L.

Abstract

Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an $L$-dimensional vector of data symbols over a base field GF($q$). Vector LNC enriches the choices of coding operations at intermediate nodes, and there is a popular conjecture on the benefit of vector LNC over scalar LNC in terms of alphabet size of data units: there exist (single-source) multicast networks that are vector linearly solvable of dimension $L$ over GF($q$) but not scalar linearly solvable over any field of size $q' \leq q^L$. This paper introduces a systematic way to construct such multicast networks, and subsequently establish explicit instances to affirm the positive answer of this conjecture for \emph{infinitely many} alphabet sizes $p^L$ with respect to an \emph{arbitrary} prime $p$. On the other hand, this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension $L$ over GF(2) but have scalar linear solutions over GF($q'$) for some $q' < 2^L$, where $q'$ can be odd or even. This discovery also unveils that over a given base field, a multicast network that has a vector linear solution of dimension $L$ does not necessarily have a vector linear solution of dimension $L' > L$.