If $X$ is a set and $A,B\subseteq X$ we let the Hamming distance of $A,B$ be defined as $d_H(A,B) = \big|(A\setminus B)\cup(B\setminus A)\big|$. If $\newcommand{\S}{{\cal S}}\S\subseteq {\cal P}(X)$, let the discrepancy of ${\cal S}$ be definied by $\newcommand{\d}{\text{discr}}\d(\S)=\min\{d_H(A,B): A\neq B\in\S\}$.

For $n\in\mathbb{N}$, let $[n]=\{1,\ldots,n\}$. Let $X$ be a set and $n\in\mathbb{N}$. Then $[X]^n$ denotes the collection of subsets of $X$ having $n$ elements.

For $n\in\mathbb{N}$, let $M_n:= \max\{\d(\S):\S\in \big[{\cal P}([n])\big]^n\}$.

Question. Is there an explicit formula for $M_n$ for all $n\in\mathbb{N}$?

EDIT. I propose as hypotheses the following "almost explicit formulas":

• $M_{2k} = k$ , and
• $M_{2k+1} \in \{k, k+1\}$

for all $k\in\mathbb{N}$ with $k\geq 2$.