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Coded caching aims to minimize the network's peak-time communication load byleveraging the information pre-stored in the local caches at the users. Theoriginal single file retrieval setting by Maddah-Ali and Niesen has beenrecently extended to general Scalar Linear Function Retrieval (SLFR) by Wan etal., who proposed a linear scheme that surprisingly achieves the same optimalload (under the constraint of uncoded cache placement) as in single fileretrieval. This paper's goal is to characterize the conditions under which ageneral SLFR linear scheme is optimal and gain practical insights into why thespecific choices made by Wan et al. work. This paper shows that the optimaldecoding coefficients are necessarily the product of two terms, one onlyinvolving the encoding coefficients and the other only the demands. Inaddition, the relationships among the encoding coefficients are shown to becaptured by the cycles of certain graphs. Thus, a general linear scheme forSLFR can be found by solving a spanning tree problem.

A General Coded Caching Scheme for Scalar Linear Function Retrieval