A FieldSize 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 sideinformation hypergraph. The sideinformation hypergraph becomes a sideinformation 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 sideinformation graph depends on the number of edges present in the sideinformation graph. In this paper, we define the notion of minrankcritical edges in a sideinformation graph and derive some properties of minrank, which identifies minranknoncritical 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 sideinformation graph by using some binary operations on the adjacency matrix of the sideinformation 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.

On the Minrank of Symmetric and Neighboring Sideinformation Index Coding Problems The length of an optimal scalar linear index code of a single unicast index coding problem (SUICP) is equal to the minrank of its sideinformation graph. A single unicast index coding problem is called symmetric neighboring and consecutive (SNC) sideinformation problem if it has K messages and K receivers, the kmathrm tmathrm h receiver R_k wanting the kmathrm tmathrm h message x_k and having the sideinformation D messages immediately after x_k and U (Dgeq U) messages immediately before x_k. Maleki, Cadambe and Jafar obtained the capacity of this SUICP(SNC) and proposed (U1)dimensional optimal length vector linear index codes by using Vandermonde matrices. However, for a b dimensional vector linear index code, the transmitter needs to wait for b realizations of each message and hence the latency introduced at the transmitter is proportional to b. For any given single unicast index coding problem with the sideinformation graph G, MAIS (G) is used to give a lowerbound on the broadcast rate of the ICP. In this paper, we analyse the properties of minrank of SUICP(SNC) sideinformation graph. We derive the MAIS (G) of sideinformation graph G of SUICP(SNC). For arbitrary K, D and U, we construct scalar linear index codes for SUICP(SNC) with length displaystyle lceil frac KU1rceil  displaystyle lfloor frac DUU1rfloor . We obtain the minrank of SUICP(SNC) sideinformation graph and show that the length of the constructed scalar linear index codes is equal to minrank of SUICP(SNC) sideinformation graph for some combinations of K, D and U.