Sets Having Special Properties Let $\mathbb{F}^{n}_{2},$ where $n>3$ be the vector space of all $n$-tuples over binary field $\mathbb{F}_{2}=\{0, 1\}$ and $\mathrm{A}\subset \mathbb{F}^{n}_{2}$ be such that $\textbf{b}+\mathrm{A}\cap \mathrm{A}\neq \emptyset $ for all $\textbf{b}\in \mathbb{F}^{n}_{2} \ \text{and}\ \lvert\mathrm{A}\rvert=2^{n-1},$ where $\textbf{b}+\mathrm{A}=\{\textbf{b}+\textbf{a} : \textbf{a}\in \mathrm{A}\}.$  Then I want to prove $$\textbf{c}+\mathrm{A}\neq \mathrm{A}\ \text{for all}\ \textbf{c}\in \mathrm{A}\ \text{where}\ \textbf{c}\neq 0, \ \text{if}\ 0\in \mathrm{A}.$$ I have verified the result for n=4 in SageMath. Please help me in this regard for general $n$. Thanks in advance.
 A: It doesn't work for $n=7$. Regard each vector as being made up of three pairs of digits plus a final digit.
Let $A'$ be the set containing every vector for which none of the three pairs is $(1,1)$.
Notice that $\dfrac{|A'|}{2^7}=\left(\dfrac{3}{4}\right)^3<\dfrac{1}{2}$.
Also, notice that $|A'|$ is even.
Finally, for all $b\in\mathbb{F}_2^7$, there exists $a_2,a_1\in A'$ such that $a_2-a_1=b$. (Exercise!)
Now, let us construct $A$ as a set containing every element of $A'$ but also adding in extra elements $\langle x,0\rangle$ and $\langle x,1\rangle$ where $x$ is supposed to represent the combination of three pairs. For each such $x$, if $A$ contains $\langle x,0\rangle$, I want $A$ to contain $\langle x,1\rangle$ as well.
Once you've added enough of these, you'll get exactly $\dfrac{|A|}{2^7}=\dfrac{1}{2}$, since $|A|$ is even.
But also, let $c$ be the vector with final digit $1$ and all other digits $0$. Observe that $c\in A$ and $c+A=A$. Also, since $A'\subset A$, for any $b\in\mathbb{F}_2^7$ you will still be able to find $a_2,a_1\in A$ such that $a_2-a_1=b$.
