  Linear index coding can be formulated as an interference alignment problem,
in which precoding vectors of the minimum possible length are to be assigned to
the messages in such a way that the precoding vector of a demand (at some
receiver) is independent of the space of the interference (non
side-information) precoding vectors. An index code has rate $\frac{1}{l}$ if
the assigned vectors are of length $l$. In this paper, we introduce the notion
of strictly rate $\frac{1}{L}$ message subsets which must necessarily be
allocated 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 for
rate $\frac{1}{3}$ feasibility using intersections of strictly rate
$\frac{1}{L}$ message subsets. We apply the necessary condition to show that
the presence of certain interference configurations makes the index coding
problem rate $\frac{1}{3}$ infeasible. We also obtain a class of index coding
problems, containing certain interference configurations, which are rate
$\frac{1}{3}$ feasible based on the idea of \textit{contractions} of an index
coding problem. Our necessary conditions for rate $\frac{1}{3}$ feasibility and
the class of rate $\frac{1}{3}$ feasible problems obtained subsume all such
known results for rate $\frac{1}{3}$ index coding.
