Optimal Error Correcting Index Codes for Some Generalized Index Coding Problems

The index coding with side-information problem introduced by Birk and Kol has been generalized to the case, where the transmissions are subjected to errors by Dau <italic>et al.</italic>. Lower and upper bounds on the optimal number of transmissions required to correct a specified number <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> of errors were established. In another generalization of index coding problem, namely, generalized index coding (GIC) problem introduced by Dai <italic>et al.</italic>, linear combination of the messages can be demanded and held as side information by the receivers. Error correction for GIC problems was studied and the bounds for optimal number of transmissions required for <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-error correcting generalized index codes were established by Byrne and Calderini. In this paper, for a particular class of GIC problems construction of optimal scalar linear index codes are discussed. For this class of GIC problems, the optimal linear <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-error correcting index codes are also constructed. As special cases, optimal linear error correcting codes are obtained for two classes of original index coding problems, namely, single-prior index coding problems and single unicast index coding problems with symmetric neighboring consecutive side information.