Optimal Scalar Linear Index Codes for Some Symmetric Multiple Unicast Problems

Abstract—The capacity of symmetric instance of the multiple unicast index coding problem with neighboring antidotes (sideinformation) with number of messages equal to the number of receivers was given by Maleki, Cadambe and Jafar in [1]. In this paper we consider ten symmetric multiple unicast problems with lesser antidotes than considered in [1] and explicitly construct scalar linear codes for them. These codes are shown to achieve the capacity or equivalently these codes shown to be of optimal length. Also, the constructed codes enable the receivers use small number of transmissions to decode their wanted messages which is important to have the probability of message error reduced in a noisy broadcast channel [8], [10]. Some of the cases considered are shown to be critical index coding problems and these codes help to identify some of the subclasses considered in [1] to be not critical index coding problems. 1