  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, $q\geq n$. 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 $n=3$.
