Bounding the Optimal Rate of the ICSI and ICCSI Problems

In this work we study both the index coding with side information (ICSI) problem introduced by Birk and Kol in 1998 and the more general problem of index coding with coded side information (ICCSI), described by Shum et al. in 2012. We estimate the optimal rate of an instance of the index coding problem. In the ICSI problem case, we characterize those digraphs having min-rank one less than their order and we give an upper bound on the min-rank of a hypergraph whose incidence matrix can be associated with that of a 2-design. Security aspects are discussed in the particular case when the design is a projective plane. For the coded side information case, we extend the graph theoretic upper bounds given by Shanmugam et al. in 2014 on the optimal rate of index code.