[IEEE Trans. on Information Theory, January 2018, pp. 274-291 ]
Linear Network Coding over Rings, Part I: Scalar Codes and Commutative Alphabets
Joseph Connelly and Kenneth Zeger
Abstract
Linear network coding over finite fields is a well-studied problem.
We consider the more general setting of linear coding
for directed acyclic networks
with finite commutative ring alphabets.
Our results imply that for scalar linear network coding over commutative rings,
fields can always be used when the alphabet size is flexible,
but other rings may be needed when the alphabet size is fixed.
We prove that if a network has a scalar linear solution over some finite commutative ring, then
the (unique) smallest such commutative ring is a field.
We also show that
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.
We study commutative rings that are maximal with respect to this quasi-order,
as they may be considered the best commutative rings of a given size.
We prove that a commutative ring is maximal 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 ring's size.
Finally, we prove 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}.
As consequences, (i) every finite field is such a maximal ring,
and (ii) for each prime p,
some network is scalar linearly solvable over a commutative ring of size pk
but not over the field of the same size
if and only if k ∉ {1,2,3,4,6}.