We prove that the sign-rank of the $k$-Hamming Distance matrix on $n$ bits is $2^{O(k)}$, independent of the number of bits $n$. This strongly refutes the conjecture of Hatami, Hatami, Pires, Tao, and Zhao (RANDOM 2022), and Hatami, Hosseini, and Meng (STOC 2023), repeated in several other papers, that the sign-rank should depend on $n$. This conjecture would have qualitatively separated margin from sign-rank (or, equivalently, bounded-error from unbounded-error randomized communication). In fact, our technique gives constant sign-rank upper bounds for all matrices which reduce to $k$-Hamming Distance, as well as large-margin matrices recently shown to be irreducible to $k$-Hamming Distance.

翻译：我们证明，$n$位$k$-汉明距离矩阵的符号秩为$2^{O(k)}$，与位数$n$无关。这一结果强有力地否定了Hatami、Hatami、Pires、Tao和Zhao（RANDOM 2022）以及Hatami、Hosseini和Meng（STOC 2023）在数篇论文中重复提出的猜想，即符号秩应依赖于$n$。该猜想本可定性地区分间隔与符号秩（或等价地，有界误差与无界误差随机通信）。事实上，我们的技术为所有可归约为$k$-汉明距离的矩阵，以及最近被证明不可归约为$k$-汉明距离的大间隔矩阵，均给出了常数符号秩上界。