Optimal Weakly Secure Linear Codes for Some Classes of the Two-Sender Index Coding Problem

The two-sender unicast index coding problem is the most fundamental multi-sender index coding problem. The two senders collectively cater to the demands of all the receivers, by taking advantage of the knowledge of their side-information. Each receiver demands a unique message and has some side-information. Weakly secure index coding problem is a practical version of the index coding problem in the presence of an eavesdropper. The eavesdropper can not gain any information about the messages he does not have, by listening to the senders' transmissions. We provide constructions of weakly secure linear codes for different classes of the two-sender unicast index coding problem, using those of its sub-problems. The constructions are valid only if such codes exist for all the sub-problems under consideration. We identify some classes of the two-sender problem, where the constructions provide optimal weakly secure linear index codes.