  The two-sender groupcast index coding problem (TGICP) consists of a set of
receivers, where all the messages demanded by the set of receivers are
distributed among the two senders. The senders can possibly have a set of
messages in common. Each message can be demanded by more than one receiver.
Each receiver has a subset of messages (known as its side information) and
demands a message it does not have. The objective is to design scalar linear
codes at the senders with the minimum aggregate code length such that all the
receivers are able to decode their demands, by leveraging the knowledge of the
side information of all the receivers. In this work, optimal scalar linear
codes of three sub-problems (considered as single-sender groupcast index coding
problems (SGICPs)) of the TGICP are used to construct optimal scalar linear
codes for some classes of the TGICP. We introduce the notion of joint
extensions of a finite number of SGICPs, which generalizes the notion of
extensions of a single SGICP introduced in a prior work. An SGICP
$\mathcal{I}_E$ is said to be a joint extension of a finite number of SGICPs if
all the SGICPs are disjoint sub-problems of $\mathcal{I}_E$. We identify a
class of joint extensions, where optimal scalar linear codes of the joint
extensions can be constructed using those of the sub-problems. We then
construct scalar linear codes for some classes of the TGICP, when one or more
sub-problems of the TGICP belong to the above identified class of joint
extensions, and provide some necessary conditions for the optimality of the
construction.
