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Linear index coding can be formulated as an interference alignment problem,in which precoding vectors of the minimum possible length are to be assigned tothe messages in such a way that the precoding vector of a demand (at somereceiver) is independent of the space of the interference (nonside-information) precoding vectors. An index code has rate $\frac{1}{l}$ ifthe assigned vectors are of length $l$. In this paper, we introduce the notionof strictly rate $\frac{1}{L}$ message subsets which must necessarily beallocated precoding vectors from a strictly $L$-dimensional space ($L=1,2,3$)in any rate $\frac{1}{3}$ code. We develop a general necessary condition forrate $\frac{1}{3}$ feasibility using intersections of strictly rate$\frac{1}{L}$ message subsets. We apply the necessary condition to show thatthe presence of certain interference configurations makes the index codingproblem rate $\frac{1}{3}$ infeasible. We also obtain a class of index codingproblems, containing certain interference configurations, which are rate$\frac{1}{3}$ feasible based on the idea of \textit{contractions} of an indexcoding problem. Our necessary conditions for rate $\frac{1}{3}$ feasibility andthe class of rate $\frac{1}{3}$ feasible problems obtained subsume all suchknown results for rate $\frac{1}{3}$ index coding.

Rate $\frac{1}{3}$ Index Coding: Forbidden and Feasible Configurations