Spanoids - an abstraction of spanning structures, and a barrier for LCCs
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We introduce a simple logical inference structure we call a spanoid (generalizing the notion of a matroid), which captures well-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 investigation of spanoids, from computational and structural viewpoints, focusing on parameters relevant to the applications areas above and, in particular, to questions regarding Locally Correctable Codes (LCCs). One central parameter we study is the rank of a spanoid, extending the rank of a matroid and related to the dimension of codes. This leads to one main application of our work, establishing the first known barrier to improving the nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. On the 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 we show that there exist (random) spanoids whose rank matches these bounds. Thus, to significantly improve the known bounds one must step out of the spanoid framework. Another parameter we explore is the 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 the main open problem we raise as it may reveal new avenues for constructing new LCCs (perhaps even matching the KT bound). As a modest first step, we develop an entropy relaxation of functional rank and use it to demonstrate a small constant gap between the two notions. To facilitate the above results we also develop some basic structural results on spanoids including an equivalent formulation of spanoids as set systems and properties of spanoid products.
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