On the Optimality of Linear Index Coding over the Fields with Characteristic Three

TL;DR

This paper investigates the limitations of linear index coding over fields with characteristic three, demonstrating instances where linear coding is suboptimal and nonlinear codes outperform them, thus extending previous results beyond characteristic two.

Contribution

It introduces new index coding instances over characteristic three fields, proving linear coding's insufficiency and showcasing nonlinear codes' advantages, with reductions to small matroid problems.

Findings

Linear coding is optimal only over certain fields with characteristic three.

Nonlinear codes can outperform linear codes in specific index coding instances.

Small matroid instances explain the limitations of linear index coding.

Abstract

It has been known that the insufficiency of linear coding in achieving the optimal rate of the general index coding problem is rooted in its rate's dependency on the field size. However, this dependency has been described only through the two well-known matroid instances, namely the Fano and non-Fano matroids, which, in turn, limits its scope only to the fields with characteristic two. In this paper, we extend this scope to demonstrate the reliance of linear index coding rate on fields with characteristic three. By constructing two index coding instances of size 29, we prove that for the first instance, linear coding is optimal only over the fields with characteristic three, and for the second instance, linear coding over any field with characteristic three can never be optimal. Then, a variation of the second instance is designed as the third index coding instance of size 58. For this instance, it is proved that while linear coding over any field with characteristic three cannot be optimal, there exists a nonlinear code over the fields with characteristic three, which achieves its optimal rate. Connecting the first and third index coding instances in two specific ways, called no-way and two-way connections, will lead to two new index coding instances of size 87 and 91, for which linear coding is outperformed by nonlinear codes. Another main contribution of this paper is the reduction of the key constraints on the space of the linear coding for the first and second index coding instances, each of size 29, into a matroid instance with the ground set of size 9, whose linear representability is dependent on the fields with characteristic three. The proofs and discussions provided in this paper through using these two relatively small matroid instances will shed light on the underlying reason causing the linear coding to become insufficient for the general index coding problem.