Upper bounds for the size of set systems with a symmetric set of Hamming distances

Let \mbox{\cal F}\subseteq 2^{[n]} be a fixed family of subsets. Let D(\mbox{\cal F}) stand for the following set of Hamming distances: D($\cal F$):=\{d_H(F,G):~ F, G\in $\cal F$,\ F\neq G\}. \mbox{\cal F} is said to be a Hamming symmetric family, if d\in D(\mbox{\cal F}) implies n-d\in D(\mbox{\cal F}) for each d\in D(\mbox{\cal F}). We give sharp upper bounds for the size of Hamming symmetric families. Our proof is based on the linear algebra bound method.