On the Overhead of Telescopic Codes in Network Coded Cooperation

Although Random Linear Network Coding (RLNC) has been shown to reduce the number of transmissions for multi-cast scenarios, there is an inherent signaling overhead to achieve this benefit. Employing a large field result in less transmissions but at the cost of a large coding vector of coefficients. Conversely, a smaller field fewer bits to represent the coding coefficients, but has the drawback of requiring more transmissions to deliver the data due to a high probability of linear dependence. This work advocates for the use of telescopic network codes as a way to achieve low overhead for cooperation in multicast transmissions over mobile networks. The idea behind them is to design a field with a larger size as a composite of a smaller field in order to be able to use some of the arithmetics of the extended field as operations in the base field, in order to allow for the use of different fields within a single generation. The resulting codes posses very low overhead by having a short coded packet representation and still maintain a low linearly dependent probability for the last packets. We provide an analytical framework and numerical results showing that is feasible to attain less than 3% total mean overhead. This is considerably lower than what can be achieved with RLNC schemes in most of the considered cases and achieving at least 1.5-2x gains.