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Groupcast Index Coding Problem: Joint Extensions
The groupcast index coding problem is the most general version of the
classical index coding problem, where any receiver can demand messages that are
also demanded by other receivers. Any groupcast index coding problem is
described by its \emph{fitting matrix} which contains unknown entries along
with 's and 's. The problem of finding an optimal scalar linear code is
equivalent to completing this matrix with known entries such that the rank of
the resulting matrix is minimized. Any row basis of such a completion gives an
optimal \emph{scalar linear} code. An index coding problem is said to be a
joint extension of a finite number of index coding problems, if the fitting
matrices of these problems are disjoint submatrices of the fitting matrix of
the jointly extended problem. In this paper, a class of joint extensions of any
finite number of groupcast index coding problems is identified, where the
relation between the fitting matrices of the sub-problems present in the
fitting matrix of the jointly extended problem is defined by a base problem. A
lower bound on the \emph{minrank} (optimal scalar linear codelength) of the
jointly extended problem is given in terms of those of the sub-problems. This
lower bound also has a dependence on the base problem and is operationally
useful in finding lower bounds of the jointly extended problems when the
minranks of all the sub-problems are known. We provide an algorithm to
construct scalar linear codes (not optimal in general), for any groupcast
problem belonging to the class of jointly extended problems identified in this
paper. The algorithm uses scalar linear codes of all the sub-problems and the
base problem. We also identify some subclasses, where the constructed codes are
scalar linear optimal.Comment: 9 page