Spanoids -生成结构的抽象，以及LCC的障碍
Spanoids - an abstraction of spanning structures, and a barrier for LCCs
摘要：
我们引入一个简单的逻辑推理结构，我们称之为一个$\textsf{spanoid}$（概括的概念，拟阵），它捕获了在几个领域的研究问题。这些包括组合几何，代数（超曲面和理想的安排），统计物理（自举渗透）和编码理论。我们开始一个彻底的调查spanoids，从计算和结构的角度来看，专注于参数相关的应用领域，特别是对局部可纠正码（LCC）的问题。我们研究的一个中心参数是spanoid的$\textsf{rank}$，扩展了拟阵的秩，并与码的维数有关。这导致了我们的工作的一个主要应用，建立了第一个已知的障碍，以改善近20年的Katz-Trevisan（KT）的限制在LCC的尺寸。一方面，我们证明了KT界（及其最近的改进）持有更一般的设置spanoid秩。另一方面，我们表明，存在（随机）spanoids的秩匹配这些界限。因此，为了显著地改进已知的边界，必须跳出Spanoid框架。我们探索的另一个参数是spanoid的$\textsf{functional rank}$，它捕获将给定的spanoid转换为实际代码的可能性。秩和函数秩之间的关系问题是我们提出的主要问题之一，因为它可能揭示构建新LCC的新途径（甚至可能匹配KT界）。作为第一步，我们开发了一个熵松弛的功能秩创建一个小的恒定的差距，并通过张量放大它，以构建一个spanoid的功能秩小于秩的多项式因子。这是证据，我们开发的熵方法可以证明多项式更好的界限比KT型方法的尺寸LCC。
英文摘要：
We introduce a simple logical inference structure we call a $\textsf{spanoid}$ (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 $\textsf{rank}$ 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 $\textsf{functional rank}$ 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 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.
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