# Capacity and Achievable Rate Regions for Linear Network Coding over Ring Alphabets

## TL;DR

This paper investigates the capacities and rate regions of linear network coding over rings and modules, revealing how these capacities depend on the algebraic properties of the alphabets and their relation to finite fields.

## Contribution

It establishes the dependence of linear capacities on the characteristic of the field and compares capacities over rings and fields, highlighting cases where rings outperform fields.

## Key findings

- Linear rate region depends only on the characteristic of the finite field.
- Capacities over rings can exceed those over fields when the characteristic does not divide the ring size.
- Networks can have higher capacities over certain rings than over finite fields.

## Abstract

The rate of a network code is the ratio of the block size of the network's messages to that of its edge codewords. We compare the linear capacities and achievable rate regions of networks using finite field alphabets to the more general cases of arbitrary ring and module alphabets. For non-commutative rings, two-sided linearity is allowed. Specifically, we prove the following for directed acyclic networks: (i) The linear rate region and the linear capacity of any network over a finite field depend only on the characteristic of the field. Furthermore, any two fields with different characteristics yield different linear capacities for at least one network. (ii) Whenever the characteristic of a given finite field divides the size of a given finite ring, each network's linear rate region over the ring is contained in its linear rate region over the field. Thus, any network's linear capacity over a field is at least its linear capacity over any other ring of the same size. An analogous result also holds for linear network codes over module alphabets. (iii) Whenever the characteristic of a given finite field does not divide the size of a given finite ring, there is some network whose linear capacity over the ring is strictly greater than its linear capacity over the field. Thus, for any finite field, there always exist rings over which some networks have higher linear capacities than over the field.