An Update-based Maximum Column Distance Coding Scheme for Index Coding

TL;DR

This paper introduces a new update-based maximum column distance (UMCD) coding scheme for index coding that improves broadcast efficiency by systematically eliminating receivers and determining optimal coding strategies through polynomial-time algorithms.

Contribution

The paper presents a novel scalar linear coding scheme for index coding, with a polynomial-time method to identify satisfied receivers and optimize transmissions, outperforming existing schemes in certain cases.

Findings

UMCD scheme outperforms recursive and ICC schemes in specific instances.

The scheme performs at least as well as MDS coding in broadcast rate.

The performance gap grows linearly with the number of messages.

Abstract

In this paper, we propose a new scalar linear coding scheme for the index coding problem called update-based maximum column distance (UMCD) coding scheme. The central idea in each transmission is to code messages such that one of the receivers with the minimum size of side information is instantaneously eliminated from unsatisfied receivers. One main contribution of the paper is to prove that the other satisfied receivers can be identified after each transmission, using a polynomial-time algorithm solving the well-known maximum cardinality matching problem in graph theory. This leads to determining the total number of transmissions without knowing the coding coefficients. Once this number and what messages to transmit in each round are found, we then propose a method to determine all coding coefficients from a sufficiently large finite field. We provide concrete instances where the proposed UMCD coding scheme has a better broadcast performance compared to the most efficient existing coding schemes, including the recursive scheme (Arbabjolfaei and Kim, 2014) and the interlinked-cycle cover (ICC) scheme (Thapa et al., 2017). We prove that the proposed UMCD coding scheme performs at least as well as the MDS coding scheme in terms of broadcast rate. By characterizing two classes of index coding instances, we show that the gap between the broadcast rates of the recursive and ICC schemes and the UMCD scheme grows linearly with the number of messages. Then, we extend the UMCD coding scheme to its vector version by applying it as a basic coding block to solve the subinstances.