Broadcast Rate Requires Nonlinear Coding in a Unicast Index Coding Instance of Size 36

TL;DR

This paper demonstrates that for a specific unicast index coding problem with 36 users, nonlinear coding outperforms linear coding in terms of broadcast rate, proving linear coding's insufficiency.

Contribution

It constructs a small unicast index coding instance where nonlinear coding surpasses linear coding, settling an open question about the necessity of nonlinear coding for the symmetric capacity.

Findings

Nonlinear coding achieves better broadcast rate than linear coding in the constructed instance.

The constructed instance has only 36 users, making the result more practical and explicit.

Linear coding is insufficient for optimal broadcast rate in certain unicast index coding problems.

Abstract

Insufficiency of linear coding for the network coding problem was first proved by providing an instance which is solvable only by nonlinear network coding (Dougherty et al., 2005).Based on the work of Effros, et al., 2015, this specific network coding instance can be modeled as a groupcast index coding (GIC)instance with 74 messages and 80 users (where a message can be requested by multiple users). This proves the insufficiency of linear coding for the GIC problem. Using the systematic approach proposed by Maleki et al., 2014, the aforementioned GIC instance can be cast into a unicast index coding (UIC) instance with more than 200 users, each wanting a unique message. This confirms the necessity of nonlinear coding for the UIC problem, but only for achieving the entire capacity region. Nevertheless, the question of whether nonlinear coding is required to achieve the symmetric capacity (broadcast rate) of the UIC problem remained open. In this paper, we settle this question and prove the insufficiency of linear coding, by directly building a UIC instance with only 36users for which there exists a nonlinear index code outperforming the optimal linear code in terms of the broadcast rate.