The $K$ User Linear Computation Broadcast (LCBC) problem is comprised of $d$ dimensional data (from $\mathbb{F}_q$), that is fully available to a central server, and $K$ users, who require various linear computations of the data, and have prior knowledge of various linear functions of the data as side-information. The optimal broadcast cost is the minimum number of $q$-ary symbols to be broadcast by the server per computation instance, for every user to retrieve its desired computation. The reciprocal of the optimal broadcast cost is called the capacity. The main contribution of this paper is the exact capacity characterization for the $K=3$ user LCBC for all cases, i.e., for arbitrary finite fields $\mathbb{F}_q$, arbitrary data dimension $d$, and arbitrary linear side-informations and demands at each user. A remarkable aspect of the converse is that unlike the $2$ user LCBC whose capacity was determined previously, the entropic formulation (where the entropies of demands and side-informations are specified, but not their functional forms) is insufficient to obtain a tight converse for the $3$ user LCBC. Instead, the converse exploits functional submodularity. Notable aspects of achievability include a decomposition of the users' collective signal space into subspaces that allow different degrees of efficiency in broadcast cost, revealing a tradeoff that leads to a constrained water-filling solution. Random coding arguments are invoked to resolve compatibility issues that arise as each user has a different view of these subspaces, conditioned on its own side-information.

翻译：$K$用户线性比较广播(LCBC)问题由中央服务器完全可用的美元维度数据($mathbb{F ⁇ q$)和K$用户组成,前者要求数据的各种线性计算,并事先了解数据作为侧信息的各种线性功能。最佳广播成本是每个计算实例由服务器播放的最起码的$q-y符号数,每个用户都可检索其预期的计算。最佳广播成本的对等性称为能力。本文的主要贡献是所有案例的K=3美元用户直径数据($K=3$用户LCBC)的准确能力描述,即任意的有限域数据计算,后者要求将数据任意的线性功能性侧信息作为侧信息,每个用户的侧面信息要求。与以前确定的2美元用户LCBCBC的计算能力不同之处不同, 最佳广播成本的对应性配方(即需求和侧面信息是指定的,但非用户LCBCLC的精确度的精确度, 其功能性格式无法通过这些功能性版本获取用户的直径直径直方数据。