Network embedding operations preserving the insufficiency of linear network codes

Operations for extending/embedding a smaller network into a larger network that preserve the insufficiency of classes of linear network codes are presented. Linear network codes over some finite field are said to be sufficient for a network if and only if for every point in the network coding rate region, there exists a code over that finite field to achieve it. Three operations are defined, and it is proven that they have the desired inheritance property, both for scalar linear network codes and for vector linear network codes, separately. Experimental results on the rate regions of multilevel diversity coding systems (MDCS), a sub-class of the broader family of multi-source multi-sink networks with special structure, are presented for demonstration. These results demonstrate that these notions of embedding operations enable one to investigate the existences of small numbers of forbidden network minors for sufficiency of linear network codes over a given field.