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.

翻译：线性编码对于网络编码问题的不足最初通过提供一个仅能由非线性网络编码求解的实例得以证明（Dougherty等，2005年）。基于Effros等人2015年的研究，该特定网络编码实例可建模为一个具有74条消息和80个用户（其中一条消息可被多个用户请求）的组播索引编码实例。这证明了线性编码对于组播索引编码问题的不足。采用Maleki等人2014年提出的系统化方法，上述组播索引编码实例可转化为一个具有200多个用户（每个用户请求唯一消息）的单播索引编码实例。这确认了非线性编码对于单播索引编码问题的必要性，但仅限于实现整个容量区域。然而，非线性编码是否为实现单播索引编码问题的对称容量（广播速率）所必需的问题仍然悬而未决。本文通过直接构建一个仅有36个用户的单播索引编码实例，证明了存在一种非线性索引码在广播速率上优于最优线性码，从而解决了该问题并证明了线性编码的不足。