Linear Network Coding over Rings, Part I: Scalar Codes and Commutative Alphabets

TL;DR

This paper explores the properties of scalar linear network coding over commutative rings, establishing conditions for maximal rings, their structure, and the relationship with fields, revealing when rings can outperform fields for fixed alphabet sizes.

Contribution

It characterizes maximal commutative rings for scalar network coding, linking their structure to prime factorization, and shows when rings outperform fields for fixed alphabet sizes.

Findings

Maximal commutative rings are characterized by prime factor multiplicities.

Unique maximal rings of size p^k are fields for k in {1,2,3,4,6}.

Fields with k not in {1,2,3,4,6} can be outperformed by other rings in network coding.

Abstract

Fixed-size commutative rings are quasi-ordered such that all scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. As consequences, if a network has a scalar linear solution over some finite commutative ring, then (i) the network is also scalar linearly solvable over a maximal commutative ring of the same size, and (ii) the (unique) smallest size commutative ring over which the network has a scalar linear solution is a field. We prove that a commutative ring is maximal with respect to the quasi-order if and only if some network is scalar linearly solvable over the ring but not over any other commutative ring of the same size. Furthermore, we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the maximal ring's size. Finally, we prove that there is a unique maximal commutative ring of size $m$ if and only if each prime factor of $m$ has multiplicity in $\{1,2,3,4,6\}$. In fact, whenever $p$ is prime and $k \in \{1,2,3,4,6\}$, the unique such maximal ring of size $p^k$ is the field $GF(p^k)$. However, for every field $GF(p^k)$ with $k\not\in \{1,2,3,4,6\}$, there is always some network that is not scalar linearly solvable over the field but is scalar linearly solvable over a commutative ring of the same size. These results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but alternative rings may be needed when the alphabet size is fixed.