Optimal Scalar Linear Index Codes for Three Classes of Two-Sender Unicast Index Coding Problem

Communication problems where a set of messages are distributed among multiple senders, can avail distributed transmissions to reduce the delay in content delivery. One such scenario is the two-sender index coding problem. 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 sub-problems (which are single-sender unicast index coding problems (SUICPs)) and the \emph{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 \emph{joint extensions} of SUICPs. An SUICP $\mathcal{I}_{E}$ is said to be a \emph{joint extension} of $l$ SUICPs if the \emph{fitting matrices} of all the $l$ SUICPs are disjoint submatrices of that of $\mathcal{I}_{E}$. Joint extensions generalize the notion of \emph{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.

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