Optimal Scalar Linear Codes for a Class of Jointly Extended Groupcast Index Coding Problems

Groupcast index coding problem is the most general version of the classical index coding problem, where any receiver can demand any number of messages and have any subset of messages as side-information as described by its fitting matrix. A single-sender index coding problem is said to be a joint extension of a set of single unicast sub-problems, if the fitting matrices of all the sub-problems are disjoint submatrices of its fitting matrix. In our prior work (C. Arunachala and B. S. Rajan, "Optimal scalar linear index codes for three classes of two-sender unicast index coding problem", International Symposium on Information Theory and its Applications (ISITA 2018)), a special class of joint extensions were used to obtain scalar linear codes for some classes of two-sender unicast index coding problems. We extend the definition of single-sender joint extensions of single-unicast sub-problems to the case, where the sub-problems can also be groupcast problems. We study a special class of such joint extensions where the extended problem is also dependent on another groupcast problem called the base problem. The fitting matrix of the base problem decides the positions of fitting matrices of the sub-problems in that of the extended problem. We then provide an algorithm to construct a scalar linear code (not optimal in general), for this class of joint extensions using scalar linear codes of all the sub-problems and the base problem. We also identify a subclass where the constructed codes are scalar linear optimal.