On the Number of Optimal Index Codes

TL;DR

This paper investigates the number of optimal index codes in the index coding problem, providing algebraic methods to determine the minimum number of such codes using network code representations.

Contribution

It introduces a simplified algebraic approach to determine the minimum number of optimal length index codes, advancing understanding of code diversity in index coding.

Findings

Derived the minimum number of optimal length index codes.

Presented an algebraic formulation for the problem.

Connected index coding with network code representations.

Abstract

In Index coding there is a single sender with multiple messages and multiple receivers each wanting a different set of messages and knowing a different set of messages a priori. The Index Coding problem is to identify the minimum number of transmissions (optimal length) to be made so that all receivers can decode their wanted messages using the transmitted symbols and their respective prior information and also the codes with optimal length. Recently it was shown that different optimal length codes perform differently in a wireless channel. Towards identifying the best optimal length index code one needs to know the number of optimal length index codes. In this paper we present results on the number of optimal length index codes making use of the representation of an index coding problem by an equivalent network code. We give the minimum number of codes possible with the optimal length. This is done using a simpler algebraic formulation of the problem compared to the approach of Koetter and Medard.