On the number of optimal linear index codes for unicast index coding problems

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 the receivers can decode their wanted messages using the transmitted symbols and their respective prior information. Recently [4], 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. Preliminary results on this have been presented in [7]. In this paper we present more results on the number of optimal linear index codes for unicast index coding problems. Specifically we obtain the exact number of optimal linear index codes for two classes of unicast index coding problems: (i) Single-unicast uniprior problems, and (ii) Single-uniprior unicast problems. For the case of single-unicast single-uniprior problems a more general proof is provided for the number of optimal linear index codes. A method to identify the optimal length codes which lead to minimum-maximum probability of error is also presented.