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

## TL;DR

This paper investigates the conditions under which rate 1/3 index coding is feasible or infeasible by analyzing interference configurations and introducing the concept of strictly rate L message subsets.

## Contribution

It introduces the notion of strictly rate L message subsets and develops necessary conditions for rate 1/3 feasibility, advancing understanding of interference configurations in index coding.

## Key findings

- Certain interference configurations make rate 1/3 infeasible.
- A class of problems with specific interference configurations are rate 1/3 feasible.
- The results generalize and unify previous findings on rate 1/3 index coding.

## Abstract

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.