Antichain Codes

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 H\'alasz's theorem, while all previously known proofs of this result are Fourier-analytic.