  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.
