Optimal linear error-correcting index codes for some generalized index coding problems

The index coding problem with side information (ICSI) was introduced by Birk and Kol, 1998. In this problem, a sender seeks to meet the demands, in minimum number of transmissions, of several receivers, each of which has some prior information on the messages (called side information). In a generalization of this problem called Generalized Index Coding problem (GICP) (Dai et al 2014), linear combination of the messages can be demanded and held as side information by the receivers. The paper by Byrne and Calderini 2017, characterizes the optimal length of linear index codes for GICP based on generalized minrank. Computation of min-rank is known to be NP-hard in general. Another generalization to ICSI problem, where transmissions are subject to noise, was addressed in Dau et al 2013 and lower and upper bounds on optimal linear ¿-error correcting index codes were established. The paper by Byrne and Calderini 2017, studied error correction for GICP and obtains lower and upper bounds on optimal linear ¿-error correcting index codes. In this paper, we find the optimal length of scalar linear index codes (min-rank) for a specific class of GICP and discuss the construction of optimal scalar linear index codes. We also find optimal error correcting index codes for some special cases of the above mentioned class and discuss the construction of optimal linear ¿-error correcting index codes.