A Field-Size Independent Code Construction for Groupcast Index Coding Problems

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 hyper-graph becomes a side-information graph for a special class of groupcast index coding problems known as unicast index coding problems. The number of computations required to find the minrank of a side-information graph 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 to reduce the number of computations required to compute minrank. Apart from this, we give a heuristic method to compute minrank. 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 give a method to construct index codes (with 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 the 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.