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We introduce a simple logical inference structure we call a$\textsf{spanoid}$ (generalizing the notion of a matroid), which captureswell-studied problems in several areas. These include combinatorial geometry,algebra (arrangements of hypersurfaces and ideals), statistical physics(bootstrap percolation) and coding theory. We initiate a thorough investigationof spanoids, from computational and structural viewpoints, focusing onparameters relevant to the applications areas above and, in particular, toquestions regarding Locally Correctable Codes (LCCs).  One central parameter we study is the $\textsf{rank}$ of a spanoid, extendingthe rank of a matroid and related to the dimension of codes. This leads to onemain application of our work, establishing the first known barrier to improvingthe nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. Onthe one hand, we prove that the KT bound (and its more recent refinements)holds for the much more general setting of spanoid rank. On the other hand weshow that there exist (random) spanoids whose rank matches these bounds. Thus,to significantly improve the known bounds one must step out of the spanoidframework.  Another parameter we explore is the $\textsf{functional rank}$ of a spanoid,which captures the possibility of turning a given spanoid into an actual code.The question of the relationship between rank and functional rank is one of themain questions we raise as it may reveal new avenues for constructing new LCCs(perhaps even matching the KT bound). As a first step, we develop an entropyrelaxation of functional rank to create a small constant gap and amplify it bytensoring to construct a spanoid whose functional rank is smaller than rank bya polynomial factor. This is evidence that the entropy method we develop canprove polynomially better bounds than KT-type methods on the dimension of LCCs.

Spanoids - an abstraction of spanning structures, and a barrier for LCCs