Abstract :

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 would 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 ANCs. As a first approximation to ANCs, we show that using infinite precision arithmetic can still lead to network codes that are computationally feasible to design and implement, and achieve the same multicast rate region as algebraic network codes. We then compute the rates achievable by ANCs for the multicast case, 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 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 size. We compare this with simulation results for randomized and deterministic design of ANCs.