论文标题
抗小节代码
Antichain Codes
论文作者
论文摘要
如果$ x \ not \ subset y $的所有不同$ x，y \ in a $ in A $中的$ x \ subset y $，则据说是一个抗小节，据说如果$ a $ a $ a $ a $ a $ hamming距离至少至少$ r $，则据说是距离$ r $ $ $。在这里，我们证明，如果$ a \ subset 2^{[n]} $既是抗小节，又是距离 - $（2r+1）$代码，则是$ | a | = o_r（2^n n^{ - r-1/2}）$。该结果最能够达到隐含常数，是纯粹的组合加强，对利特伍德理论中的许多结果进行了加强。例如，我们的结果给出了Hálasz定理的简短组合证明，而以前所有已知结果的证明都是傅立叶分析。
A family of sets $A$ is said to be an antichain if $x\not\subset y$ for all distinct $x,y\in A$, and it is said to be a distance-$r$ code if every pair of distinct elements of $A$ has Hamming distance at least $r$. Here, we prove that if $A\subset 2^{[n]}$ is both an antichain and a distance-$(2r+1)$ code, then $|A| = O_r(2^n n^{-r-1/2})$. This result, which is best-possible up to the implied constant, is a purely combinatorial strengthening of a number of results in Littlewood--Offord theory; for example, our result gives a short combinatorial proof of Hálasz's theorem, while all previously known proofs of this result are Fourier-analytic.
