AI 中文总结
该研究证明了沃尔什-哈达玛变换矩阵在F₃域上的超对数秩刚性，得到首个显式矩阵族在超对数目标秩下的常数分数刚性下界，推进了Razborov通信复杂性下界方案的参数。
AI 中文摘要
对于足够大的2的幂次N，我们证明：将N×N沃尔什-哈达玛变换的矩阵中最多1%的元素修改，无法使其在F₃域上的秩降至⌊log²N/80⌋或更低。据我们所知，这是首个在任意域上，针对显式矩阵族在超对数目标秩下的常数分数刚性下界，向Razborov通信复杂性下界方案的参数推进了一步。
英文摘要
For sufficiently large $N$ which is a power of 2, we prove that changing at most one percent of the entries of the $N\times N$ Walsh-Hadamard Transform cannot reduce its rank over $\mathbb{F}_3$ to $\lfloor \log^2 N/80\rfloor$ or below. To the best of our knowledge, this is the first constant-fraction rigidity lower bound for an explicit matrix family at a superlogarithmic target rank over any choice of field, inching toward the parameters in Razborov's program for communication complexity lower bounds.
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