Capacity of Symmetric Index Coding Problems With X-Network Setting With <inline-formula> <tex-math notation="LaTeX">$2\times2$ </tex-math> </inline-formula> Local Connectivity

Abstract

In $X$ -network setting with ${L \times L}$ local connectivity, we have a locally connected network where each receiver is connected to $L$ consecutive base stations and each base station has a distinct message for each connected receiver. Maleki et al. modeled the ${X}$ -network setting with ${L \times L}$ local connectivity as a multiple unicast index coding problem and proved that the capacity of symmetric index coding problems with locally connected ${X}$ -network with ${K}$ number of receivers and number of messages ${M=KL}$ and ${K}$ tending to infinity is ${({{2}}/[{{L(L+1)}}]})$ and for finite number of receivers this was shown to be an upper bound on the capacity. In this letter: 1) we show that when ${L=2}$ the upper bound is exact, i.e., the capacity is (1/3) and 2) for this case we give an explicit construction of optimal linear index codes to achieve this capacity by using interference alignment.