Obtaining an antichain from affine subspace

Question

Suppose $a\in \{0,1\}^n$ and $S \subseteq \mathbb{F}_2^n$ is a subspace of dimension $d$. Define an affine subspace $S_a$ as follows: $$S_a=\{a+x \mid x\in S\}.$$ What is the largest possible size of the antichain sitting inside $S_a$ for all $a \in \{0,1\}^n$?

For example, $S=\{000,001,010,011\}$. Clearly $\operatorname{dimension}(S)=2$. Observe that there is no point in choosing $a \in S$ as we will obtain the same set $S$ (by definition).

Suppose $a\in \{100,101,110,111\}$. Then affine subspace, $S_a=\{100,101,110,111\}$ irrespective of which $a$ we chose. In this scenario, the size of the antichain is 2 ($\{101,110\}$) which is same as antichain of $S$. I am wondering, if this is the case for all subspaces $S$ or we can obtain a bigger antichain. Is there an explanation for this observation?

Answer 1

Let $S=\{00,11\}$ and $a=10$. The largest antichain in $S$ has size 1, while the largest antichain in $S_a$ has size 2. We can extend this construction to get an affine subspace of any dimension $k$, all of whose elements form an antichain (of size $2^k$). Namely, letting $0^j$ denote a string of $j$ $0$'s, let $S$ be generated by $110^{2k-2}$, $00110^{2k-4}$, $0^4110^{2k-6}, \dots, 0^{2k-2}11$, and let $a=101010\cdots 10$.

Addendum. We can also ask for the size $A(k)$ of the largest possible antichain contained in a $k$-dimensional linear subspace. Clearly $A(k)\leq 2^k-1$, since the zero vector is in the subspace. The Hamming single-error correcting code $H(2^k,2^k-k-1)$ of length $n=2^k-1$ and dimension $k$ has all its nonzero vectors having $2^{k-1}$ 1's, so they form an antichain. Hence $A(k)=2^k-1$.