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

Let \( \mathcal{F} \subseteq 2^{[n]}\) be a fixed family of subsets. Let \(D( \mathcal{F} )\) stand for the following set of Hamming distances:

$$D( \mathcal{F} ):=\{d_H(F,G) : F, G\in \mathcal{F} ,\ F\neq G\}$$

. \( \mathcal{F} \) is said to be a Hamming symmetric family, if \( \mathcal{F} \)X implies \(n-d\in D( \mathcal{F} )\) for each \(d\in D( \mathcal{F} )\).

We give sharp upper bounds for the size of Hamming symmetric families. Our proof is based on the linear algebra bound method.