设计矩阵的秩界及其在组合几何和局部可纠正码中的应用
Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes
摘要：
（q，k，t）-设计矩阵是一个m × n矩阵，其零/非零模式满足以下类似设计的条件：每行最多有q个非零，每列至少有k个非零，每两列的支撑最多相交于t行。我们证明了特征为零（或充分大的有限特征）的域上的任何（q，k，t）-设计矩阵的秩至少为n -（qtn/2k）^2。（1）复数域上2-查询局部可纠正码的不可能性结果：2-查询局部可纠正码（LCC）是一种纠错码，其中每个码字的坐标都可以通过读取至多两个其他码位来恢复。这样的代码有许多应用和结构（具有指数编码长度）是已知的小特征的有限域。我们表明，这样的线性2-查询LCC的无限家庭不存在的复数。(2)推广的结果在组合几何：我们证明了一个定量模拟的Sylvester-Gallai定理：让$v_1,...,v_m$是一组点在$\C^d$，使每一个$i \in [m]$至少存在$δm$值的$j \in [m]$，这样的线通过$v_i,v_j$包含第三个点的集合。证明了$\{v_1,...,v_m \}$的维数至多为$O(1/δ^2)$ .我们的结果推广到高维情形（用平面代替直线等）。以及点被着色的情况（如Motzkin-Rabin定理）。
英文摘要：
A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank of any (q,k,t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n - (qtn/2k)^2 . Using this result we derive the following applications: (1) Impossibility results for 2-query LCCs over the complex numbers: A 2-query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are known over finite fields of small characteristic. We show that infinite families of such linear 2-query LCCs do not exist over the complex numbers. (2) Generalization of results in combinatorial geometry: We prove a quantitative analog of the Sylvester-Gallai theorem: Let $v_1,...,v_m$ be a set of points in $\C^d$ such that for every $i \in [m]$ there exists at least $δm$ values of $j \in [m]$ such that the line through $v_i,v_j$ contains a third point in the set. We show that the dimension of $\{v_1,...,v_m \}$ is at most $O(1/δ^2)$. Our results generalize to the high dimensional case (replacing lines with planes, etc.) and to the case where the points are colored (as in the Motzkin-Rabin Theorem).
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