"Real" and "Complex" Network Codes: Promises and Challenges

As an alternative to the algebraic network codes prevalent in the literature, we consider Arithmetic Network Codes (henceforth abbreviated as ANCs), i.e., codes in which interior nodes perform finite precision arithmetic over the real or complex fields. We suggest two applications where using such codes can be advantageous. First, we demonstrate that the multi-resolution behaviour of ANCs potentially outperforms that of algebraic network codes. Second, the interfering and fading nature of wireless channels naturally results in complex linear combinations of transmissions, analogous to ANCs. We then characterize the multicast rates achievable by ANCs, and demonstrate that for high precision arithmetic these are equivalent to those obtained by algebraic network codes. We show the connection between the performance of ANCs and the numerical conditioning of network transform matrices. Using this, we obtain upper and lower bounds on the number of significant bits required to perform the finite precision arithmetic in terms of the network parameters. We compare this with simulation results for randomized and deterministic design of ANCs.