# Task-based Solutions to Embedded Index Coding

## TL;DR

This paper investigates the limitations of task-based linear embedded index coding, demonstrating that for some cases, task-based codes are quadratically longer than optimal codes, highlighting fundamental bounds in the model.

## Contribution

It provides the first explicit construction showing a quadratic gap between optimal and task-based linear embedded index codes, using spectral analysis techniques.

## Key findings

- Quadratic gap between task-based and optimal embedded index codes for certain side information.
- Explicit construction demonstrating the fundamental limits of task-based restrictions.
- Spectral techniques used to analyze and prove the bounds.

## Abstract

In the index coding problem a sender holds a message $x \in \{0,1\}^n$ and wishes to broadcast information to $n$ receivers in a way that enables the $i$th receiver to retrieve the $i$th bit $x_i$. Every receiver has prior side information comprising a subset of the bits of $x$, and the goal is to minimize the length of the information sent via the broadcast channel. Porter and Wootters have recently introduced the model of embedded index coding, where the receivers also play the role of the sender and the goal is to minimize the total length of their broadcast information. An embedded index code is said to be task-based if every receiver retrieves its bit based only on the information provided by one of the receivers. This paper studies the effect of the task-based restriction on linear embedded index coding. It is shown that for certain side information maps there exists a linear embedded index code of length quadratically smaller than that of any task-based embedded index code. The result attains, up to a multiplicative constant, the largest possible gap between the two quantities. The proof is by an explicit construction and the analysis involves spectral techniques.