# Reduced Complexity Index Codes and Improved Upperbound on Broadcast Rate for Neighboring Interference Problems

## TL;DR

This paper presents a method to reduce the complexity of index coding for neighboring interference problems by partitioning messages, achieving the same rate as previous methods but with less storage requirement at receivers, and also provides an improved upper bound on broadcast rate.

## Contribution

It introduces a message partitioning technique for SUICP(SNI) that reduces encoding matrix size while maintaining optimal rate, and offers a new upper bound on broadcast rate.

## Key findings

- Reduced encoding matrix size through message partitioning.
- Receivers store fewer broadcast symbols for decoding.
- Established an improved upper bound on broadcast rate.

## Abstract

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has $K$ messages and $K$ receivers, the $k$th receiver $R_{k}$ wanting the $k$th message $x_{k}$ and having the interference with $D$ messages after and $U$ messages before its desired message. Maleki, Cadambe and Jafar studied SUICP(SNI) because of its importance in topological interference management problems. Maleki \textit{et. al.} derived the lowerbound on the broadcast rate of this setting to be $D+1$. In our earlier work, for SUICP(SNI) with arbitrary $K,D$ and $U$, we defined set $\mathbf{S}$ of 2-tuples and for every $(a,b) \in \mathbf{S}$, we constructed $b$-dimensional vector linear index code with rate $D+1+\frac{a}{b}$ by using an encoding matrix of dimension $Kb \times (b(D+1)+a)$. In this paper, we use the symmetric structure of the SUICP(SNI) to reduce the size of encoding matrix by partitioning the message symbols. The rate achieved in this paper is same as that of the existing constructions of vector linear index codes. More specifically, we construct $b$-dimensional vector linear index codes for SUICP(SNI) by partitioning the $Kb$ messages into $b(U+1)+c$ sets for some non-negative integer $c$. We use an encoding matrix of size $\frac{Kb}{b(U+1)+c} \times \frac{b(D+1)+a}{b(U+1)+c}$ to encode each partition separately. The advantage of this method is that the receivers need to store atmost $\frac{b(D+1)+a}{b(U+1)+c}$ number of broadcast symbols (index code symbols) to decode a given wanted message symbol. We also give a construction of scalar linear index codes for SUICP(SNI) with arbitrary $K,D$ and $U$. We give an improved upperbound on the braodcast rate of SUICP(SNI).