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Optimal Vector Linear Index Codes for Some Symmetric Side Information Problems

This paper deals with vector linear index codes for multiple unicast index coding problems where there is a source with K messages and there are K receivers each wanting a unique message and having symmetric (with respect to the receiver index) two-sided antidotes (side information). Optimal scalar linear index codes for several such instances of this class of problems for one-sided antidotes(not necessarily adjacent) have already been reported. These codes can be viewed as special cases of the symmetric unicast index coding problems discussed by Maleki, Cadambe and Jafar with one sided adjacent antidotes. In this paper, starting from a given multiple unicast index coding problem with with K messages and one-sided adjacent antidotes for which a scalar linear index code $\mathfrak{C}$ is known, we give a construction procedure which constructs a sequence (indexed by m) of multiple unicast index problems with two-sided adjacent antidotes (for the same source) for all of which a vector linear code $\mathfrak{C}^{(m)}$ is obtained from $\mathfrak{C}.$ Also, it is shown that if $\mathfrak{C}$ is optimal then $\mathfrak{C}^{(m)}$ is also optimal for all $m.$ We illustrate our construction for some of the known optimal scalar linear codes.