We consider a graphical approach to index coding. While cycles have been shown to provide coding gain, only disjoint cycles and cliques (a specific type of overlapping cycles) have been exploited in 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 non-linear 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 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. Index coding problem, unicast, linear index codes, interlinked-cycle cover, optimal broadcast rate.
Recently, Thapa et al. REF introduced the interlinked-cycle structure (a general form of overlapping cycles) and proposed a interlinkedcycle-cover (ICC) scheme using scalar linear index codes.
Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques
IEEE Transactions on Information Theory ( Volume: 63, Issue: 6, June 2017 )
Mathematics Computer Science