  An index coding problem arises when there is a single source with a number of
messages and multiple receivers each wanting a subset of messages and knowing a
different set of messages a priori. The noiseless Index Coding Problem is to
identify the minimum number of transmissions (optimal length) to be made by the
source through noiseless channels so that all receivers can decode their wanted
messages using the transmitted symbols and their respective prior information.
Recently, it is shown that different optimal length codes perform differently
in a noisy channel. Towards identifying the best optimal length index code one
needs to know the number of optimal length index codes. In this paper we
present results on the number of optimal length index codes making use of the
representation of an index coding problem by an equivalent network code. Our
formulation results in matrices of smaller sizes compared to the approach of
Kotter and Medard. Our formulation leads to a lower bound on the minimum number
of optimal length codes possible for all unicast index coding problems which is
met with equality for several special cases of the unicast index coding
problem. A method to identify the optimal length codes which lead to
minimum-maximum probability of error is also presented.
