Linear Capacity of Networks over Ring Alphabets

The rate of a network code is the ratio of the block sizes of the network’s messages and its edge codewords. The linear capacity of a network over a finite ring alphabet is the supremum of achievable rates using linear codes over the ring. We prove the following for directed acyclic networks: (i) For every finite field F and every finite ring R, there is some network whose linear capacity over R is greater than over F if and only if the sizes of F and R are relatively prime. (ii) Every network’s linear capacity over a finite ring is less than or equal to its linear capacity over every finite field whose characteristic divides the ring’s size. As a consequence, every network’s linear capacity over a finite field is greater than or equal to its linear capacity over every ring of the same size. Additionally, every network’s linear capacity over a module is at most its linear capacity over some finite field. (iii) The linear capacity of any network over a finite field depends only on the characteristic of the field. (iv) Every network that is asymptotically linearly solvable over some finite ring (or even some module) is also asymptotically linearly solvable over some finite field. These results establish the sufficiency of finite field alphabets for linear network coding. Namely, linear network coding capacity cannot be improved by looking beyond finite field alphabets to more general ring alphabets. However, certain rings can yield higher linear capacities for certain networks than can a given field. ∗This work was supported by the National Science Foundation. J. Connelly and K. Zeger are with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0407 ([email protected] and [email protected]).