Bounding the optimal rate of the ICSI and ICCSI problem

TL;DR

This paper investigates the optimal transmission rates for index coding problems with side information, providing bounds and characterizations for specific cases like digraphs, hypergraphs, and projective planes, advancing understanding of index coding efficiency.

Contribution

It introduces new bounds and characterizations for the optimal rate of ICSI and ICCSI problems, including digraphs with specific min-rank properties and hypergraph incidence matrices.

Findings

Characterization of digraphs with min-rank one less than their order

Upper bounds on min-rank of hypergraphs related to 2-designs

Analysis of security aspects in projective plane designs

Abstract

In this work we study both the index coding with side information (ICSI) problem introduced by Birk and Kol in 1998 and the more general problem of index coding with coded side information (ICCSI), described by Shum et al in 2012. We estimate the optimal rate of an instance of the index coding problem. In the ICSI problem case, we characterize those digraphs having min-rank one less than their order and we give an upper bound on the min-rank of a hypergraph whose incidence matrix can be associated with that of a 2-design. Security aspects are discussed in the particular case when the design is a projective plane. For the coded side information case, we extend the graph theoretic upper bounds given by Shanmugam et al in 2014 on the optimal rate of index code.