We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called cache) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. One of the chief drawbacks of many of these existing constructions is the large subpacketization level, which denotes the number of times a file should be split for the schemes to provide coding gain. In this work, using a new binary matrix model, we present several novel constructions for coded caching based on the various types of combinatorial designs and their $q$-analogs, which are also called subspace designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement, they provide a rate that is either constant or decreasing, and moreover require competitively small levels of subpacketization, which is an extremely important feature in practical applications of coded caching. We also apply our constructions to the distributed computing framework of MapReduce, which consists of three phases, the Map phase, the Shuffle phase and the Reduce phase. Using our binary matrix framework, we present a new simple generic coded data shuffling scheme. Employing our designs-based constructions in conjunction with this new shuffling scheme, we obtain new coded computing schemes which have low file complexity, with marginally higher communication load compared to the optimal scheme for equivalent parameters. We show that our schemes can neatly extend to the scenario with full and partial stragglers also.

翻译：我们考虑标准广播场景：单个服务器向多个客户端广播信息，每个客户端拥有一定容量的本地存储（称为缓存），用于存储服务器端可用文件的部分内容。集中式编码缓存框架包含缓存阶段和交付阶段，这两个阶段均需精心设计，以便协同优化利用缓存与信道。现有文献中，已采用多种组合结构来构建编码缓存方案。现有构造的主要缺陷之一在于子包化水平过高——该指标表示为实现编码增益需将文件分割成的子块数量。本文通过引入新型二进制矩阵模型，基于各类组合设计及其$q$-模拟（亦称子空间设计），提出若干新颖的编码缓存构造。尽管本文基于现有设计的大多数构造方案对缓存容量要求较高，但它们能实现恒定或递减的传输速率，并且所需子包化水平极具竞争力，这一特性在编码缓存实际应用中至关重要。我们还将这些构造应用于MapReduce分布式计算框架（包含映射阶段、混洗阶段和归约阶段）。通过二进制矩阵框架，我们提出了一种简单通用的新型编码数据混洗方案。将该设计方案与新混洗方案结合，我们获得的新型编码计算方案具有较低的文件复杂度，同时其通信负载相较于等参数最优方案仅有轻微增加。研究表明，该方案可自然扩展至全迟滞节点与部分迟滞节点共存的场景。