Throughput Bound of XOR Coded Wireless Multicasting to Three Clients

It is a well-known result that constructing codewords over $GF(2)$ to minimize the number of transmissions for a single-hop wireless multicasting is an NP-complete problem. Linearly independent codewords can be constructed in polynomial time for all the $n$ clients, known as maximum distance separable (MDS) code, when the finite field size $q$ is larger than or equal to the number of clients, $q\ge n$. In this paper we quantify the exact minimum number of transmissions for a multicast network using erasure code when $q=2$ and $n=3$, such that $q<n$. We first show that the use of Markov chain model to derive the minimum number of transmissions for such a network is limited for very small number of input packets. We then use combinatorial approach to derive an upper bound on the exact minimum number of transmissions. Our results show that the difference between the expected number of transmissions using XOR coding and MDS coding is negligible for $n=3$.