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The index coding problem has been generalized recently to accommodatereceivers which demand functions of messages and which possess functions ofmessages. The connections between index coding and matroid theory have beenwell studied in the recent past. Index coding solutions were first connected tomulti linear representation of matroids. For vector linear index codes discretepolymatroids which can be viewed as a generalization of the matroids was used.It was shown that a vector linear solution to an index coding problem exists ifand only if there exists a representable discrete polymatroid satisfyingcertain conditions. In this work we explore the connections between generalizedindex coding and discrete polymatroids. The conditions that needs to besatisfied by a representable discrete polymatroid for a generalized indexcoding problem to have a vector linear solution is established. From a discretepolymatroid we construct an index coding problem with coded side informationand shows that if the index coding problem has a certain optimal lengthsolution then the discrete polymatroid satisfies certain properties. From amatroid we construct a similar generalized index coding problem and shows thatthe index coding problem has a binary scalar linear solution of optimal lengthif and only if the matroid is binary representable.

Generalized Index Coding Problem and Discrete Polymatroids