# Linear Network Coding: Effects of Varying the Message Dimension on the Set of Characteristics

## TL;DR

This paper investigates how changing message dimensions affects the set of finite field characteristics over which vector linear solutions exist in network coding, revealing surprising effects of increasing or decreasing message dimensions.

## Contribution

It demonstrates that altering message dimensions by one can significantly expand the set of characteristics for which solutions exist, and compares the effectiveness of rings versus finite fields.

## Key findings

- Increasing message dimension by 1 can enlarge the set of characteristics for solutions.
- Decreasing message dimension by 1 can also enlarge the set of characteristics.
- Existence of solutions at different dimensions over the same field is interconnected.

## Abstract

It is known a vector linear solution may exist if and only if the characteristic of the finite field belongs to a certain set of primes. But, can increasing the message dimension make a network vector linearly solvable over a larger set of characteristics? To the best of our knowledge, there exists no network in the literature which has a vector linear solution for some message dimension if and only if the characteristic of the finite field belongs to a set $P$, and for some other message dimension it has a vector linear solution over some finite field whose characteristic does not belong to $P$. We have found that by \textit{increasing} the message dimension just by $1$, the set of characteristics over which a vector linear solution exists may get arbitrarily larger. However, somewhat surprisingly, we have also found that by \textit{decreasing} the message dimension just by $1$, the set of characteristics over which a vector linear solution exists may get arbitrarily larger. As a consequence of these finding, we prove two more results: (i) rings may be superior to finite fields in terms of achieving a scalar linear solution over a lesser sized alphabet, (ii) existences of $m_1$ and $m_2$ dimensional vector linear solutions guarantees the existence of an $(m_1 + m_2)$-dimensional vector linear solution only if the $m_1$ and $m_2$ dimensional vector linear solutions exist over the same finite field.