Optimal Scalar Linear Index Codes for Some Two-Sender Unicast Index Coding Problems

In this paper, two-sender unicast index coding problem (TUICP) is studied, where the senders possibly have some messages in common, and each receiver requests a unique message. It is analyzed using three independent subproblems (which are single-sender unicast index coding problems (SUICPs)) and the interactions among them. These sub-problems are described by three disjoint vertex-induced subgraphs of the side-information graph of the TUICP respectively, based on the availability of messages at the senders. The TUICP is classified based on the type of interactions among the sub-problems. Optimal scalar linear index codes for a class of TUICP are obtained using those of the sub-problems. For two classes, we identify a sub-class for which scalar linear codes are obtained using the notion of joint extensions of SUICPs. An SUICP I
_E
is said to be a joint extension of l SUICPs if the fitting matrices of all the l SUICPs are disjoint submatrices of that of I
_E
. Joint extensions generalize the notion of rank-invariant extensions. Scalar linear codes and a condition for optimality of the codes are given for a class of joint extensions. Using this result, scalar linear codes and the conditions for their optimality are obtained for two classes of the TUICP.