SIAM Journal on Computing, Volume 49, Issue 3, Page 465-496, January 2020.
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 (point-line incidences), algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation), network theory (gossip/infection processes) 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 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 entropy relaxation of functional rank to create a small constant gap and amplify it by tensoring to construct a spanoid whose functional rank is smaller than rank by a polynomial factor. This is evidence that the entropy method we develop can prove polynomially better bounds than KT-type methods on the dimension of LCCs. 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. We feel that given these initial findings and their motivations, the abstract study of spanoids merits further investigation. We leave plenty of concrete open problems and directions.

中文翻译：

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Spanoids--跨接结构的抽象和LCC的障碍

SIAM计算学报，第49卷，第3期，第465-496页，2020年1月。
我们介绍了一种简单的逻辑推理结构，我们称其为“ spanoid”（泛化了类人动物的概念），它捕获了多个领域中经过充分研究的问题，包括组合几何（点线入射），代数（超曲面的排列和理想值） ），统计物理学（自举渗透），网络理论（八卦/感染过程）和编码理论。我们从计算和结构的角度出发，对跨距进行了彻底的研究，重点是与上述应用领域相关的参数，尤其是有关局部可纠正代码（LCC）的问题，我们研究的一个中心参数是螺线的“等级”，它扩展了拟阵的等级，并与代码的尺寸有关。这导致了我们工作的一个主要应用，建立了第一个已知的障碍，以改善LCC规模近20年的Katz-Trevisan（KT）界限。一方面，我们证明了KT界线（及其最近的改进）适用于更广泛的spanoid rank设置。另一方面，我们表明存在（随机）的类凸肌，其等级与这些范围匹配。因此，要显着改善已知界限，必须走出跨距框架。我们探索的另一个参数是spanoid的“功能等级”，它捕获了将给定的spanoid转换为实际代码的可能性，等级和功能等级之间的关系是我们提出的主要问题之一构造新的LCC的新途径（可能甚至与KT边界匹配）。第一步，我们开发了一个功能等级的熵松弛来创建一个小的恒定间隙，并通过张量法构造一个功能等级小于等级乘以多项式因子的斯班尼放大它。这证明了我们开发的熵方法在LCC的尺寸上可以证明比KT型方法更好的多项式边界。为了促进上述结果，我们还开发了一些关于spanoids的基本结构结果，包括将spanoids等效地设置为set制度和spanoid产品的特性。我们认为，鉴于这些最初的发现及其动机，对类动物的抽象研究值得进一步研究。我们留下了大量具体的公开问题和方向。