Characterizing the Number of Optimal Index Codes and Their Relation to Min-Max Probability of Error Performance

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 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 a noiseless channel so that all receivers can decode their wanted messages using the transmitted symbols and their respective prior information. Recently (A. Thomas, R. Kavitha, A. Chandramouli, and B. Sundar Rajan, ``Optimal index coding with min-max probability of error over fading channels,'' in Proc. IEEE 26th Annu. Int. Symp. Pers. Indoor Mobile Radio Commun., Hong Kong, 2015, pp. 889-894), it is shown that different optimal length codes perform differently in a noisy channel for single-uniprior index coding problems. Towards identifying the best optimal length index code, one needs to know the number of optimal length index codes. In this paper, we consider unicast index coding problems and 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 Koetter and Medard (R. Koetter and M. Medard, ``An algebraic approach to network coding,'' IEEE/ACM Trans. Netw., vol. 11, no. 5, pp. 782-795, Oct. 2003.). Our formulation leads to a lower bound on the minimum number of optimal length codes possible for all unicast index coding problems (L. Ong and C. K. Ho, ``Optimal index codes for a class of multicast networks with receiver side information,'' in Proc. IEEE Int. Conf. Commun., 2012, pp. 2213-2218), which is met with equality for several special cases of the unicast index coding problem. Furthermore, for noisy broadcast channels, a method to identify the best among the optimal length codes for unicast index coding problems in terms of minimum-maximum error probability is presented