A Field-Size Independent Code Construction for Groupcast Index Coding Problems
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The length of an optimal scalar linear index code of a groupcast index coding problem is equal to the minrank of its side information hypergraph. The side-information hypergraph becomes a side-information graph for a special class of groupcast index coding problems known as unicast index coding problems. The computation of minrank is an NP-hard problem. There exists a low rank matrix completion method and clique cover method to find suboptimal solutions to the index coding problem represented by a side-information graph. However, both the methods are NP-hard. The number of computations required to find the minrank depends on the number of edges present in the side-information graph. In this paper, we define the notion of minrank-critical edges in a side-information graph and derive some properties of minrank, which identifies minrank-non-critical edges. Using these properties we present a method for reduction of the given minrank computation problem into a smaller problem. Also, we give an heuristic algorithm to find a clique cover of the side-information graph by using some binary operations on the adjacency matrix of the side-information graph. We also give a method to convert a groupcast index coding problem into a single unicast index coding problem. Combining all these results, we construct index codes (not necessarily optimal length) for groupcast index coding problems. The construction technique is independent of field size and hence can be used to construct index codes over binary field. In some cases the constructed index codes are better than the best known in the literature both in terms of the length of the code and the minimum field size required.
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