Linear Network Coding Over Rings – Part I: Scalar Codes and Commutative Alphabets

Linear network coding over finite fields is a wellstudied 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 the 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 quasiorder, 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 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}. As consequences, 1) every finite field is such a maximal ring and 2) 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}.