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 et al.. Lower and upper bounds on the optimal number of transmissions required to correct a specified number δ of errors were established. In another generalization of index coding problem, namely, generalized index coding (GIC) problem introduced by Dai et al., 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 δ-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 δ-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.

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.

In an index coding with side-information (ICSI) problem there is a sender with a set of n independent messages ℳ = {x1, x2, ..., xn} and there are m receivers ℛ = {R1, R2,..., Rm}, each identified with (Wi, Ki), where Wi⊆ ℳ is the set of messages wanted by receiver Riand Ki⊂ ℳ is the set of messages known a priori to receiver Ri. We call a ICSI problem to be single-prior if |Ki| = 1, ∀i. In addition, if Ki∩ Ki= φ, it is called a single-uniprior problem and has been studied in detail by Ong, Ho and Lim [1]. Error Correcting index codes (ECIC) have been studied by Dau, Skachek and Chee [2] and lower and upper bounds on the optimal length of ECICs have been reported. In this paper we show that for single-prior ICSI problems the lower and upper bounds on the optimal ECICs coincide. This leads to construction of optimal length linear ECICs.