Interlinked cycles for index coding: generalizing cycles and cliques

We consider a graphical approach to index coding. As cycles have been shown to provide coding gain, cycles and cliques (a specific type of overlapping cycles) have been exploited in an existing literature. In this paper, we define a more general form of overlapping cycles, called the interlinked-cycle (IC) structure, that generalizes cycles and cliques. We propose a scheme, called the interlinked-cycle-cover (ICC) scheme, that leverages IC structures in digraphs to construct scalar linear index codes. We characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and nonlinear index codes. Consequently, for this class of digraphs, we indirectly prove that scalar linear index codes are optimal. Furthermore, we show that the ICC scheme can outperform all the existing graph-based schemes (including partial-clique-cover and fractional-local-chromatic number schemes), and a random coding scheme (namely, composite coding) for certain graphs.