**Definition 1.** For each $n\in\mathbb{Z}^+$, the $n$-dimensional **Hamming cube** is the set of ordered $n$-tuples of $\lbrace 0,1\rbrace$, denoted by $\lbrace 0,1\rbrace ^n$.


**Definition 2.** The binary operation that turns $\lbrace 0,1\rbrace ^n$ into a group is $\oplus$ (**XOR**), which is bitwise addition reduced modulo $2$.


**Definition 3.** The sum of the digits of an element of $\lbrace 0,1\rbrace ^n$ is its **Hamming weight**. The Hamming weight of the XOR of two elements is the **Hamming distance** $h$ between them. In fact, $h$ is a metric on $\lbrace 0,1\rbrace ^n$.


**Question.** Given integers $0\le a\le b\le n$, what is known regarding **maximal subsets** $S$ of $\lbrace 0,1\rbrace^n$ such that: $\forall x,y\in S: a\le h(x,y)\le b$ ? Specifically, is anything known about bounds on **cardinality**, or exact cardinality, or even **construction** of these maximal sets?


I expect that the full answer is not known but I would be happy to read replies about, or receive direction toward resources for, special cases. For example, if $n$ is even and if $a=b=\frac{n}{2}$, then we turn to Hadamard matrices.



Bounded Hamming distance
I think you're assuming $x \not= y$ when you say "for any $x, y \in S$." In any case, your question seems like a mix of coding theory and design theory.


If you find the case when $a = b = \frac{n}{2}$ interesting, a $q$-ary code $\mathcal{C} \subset \mathbb{F}\_q^n$ is said to be *equidistant* if for any pair $\boldsymbol{c}, \boldsymbol{c}' \in \mathcal{C}$, $\boldsymbol{c}\not=\boldsymbol{c}'$, of distinct codewords, the Hamming distance $h(\boldsymbol{c}, \boldsymbol{c}')$ is $d$. Your example is the special case when the code is binary and of equidistance $d = \frac{n}{2} (= a = b)$. We focus on the binary case for the moment.


Any equidistant code $\mathcal{C}$ can be transformed into a *constant-weight* equidistant code $\mathcal{D} \subset \mathbb{F}\_2^n$ of size $\vert \mathcal{C} \vert -1$, where $\operatorname{wt}(\boldsymbol{d}) = \operatorname{wt}(\boldsymbol{d}')$ for any $\boldsymbol{d}, \boldsymbol{d}' \in \mathcal{D}$ (see the references given below (or some references therein) for the proof if you need it). Hence, essentially we only need to consider the case when a code is both constant-weight and equidistant. Let $\operatorname{ex}(n,d,w)$ be the number of codewords of a largest binary code of lenght $n$, equidistance $d$, and constant-weight $w$ (i.e., no other constant-weight, equidistant code of the same parameters has more codewords). Then the classical result by Stinson and van Rees states that


$$\operatorname{ex}(4k+1,2k,2k) \leq 4k$$ with equality if and only if $k=\frac{u^2+u}{2}$ for some integer $u$ if there exists a $(2u^2+2u+1, u^2, \frac{u^2-u}{2})$ symmetric balanced incomplete block design (BIBD). They also proved that


$$\operatorname{ex}(4k+1,2k,2k) \geq 4k-1$$ if there exists a $(4k-1,2k,k)$-SBIBD. If you don't know what symmetric designs are, you can find the definition in your favorite design theory textbook or, better yet, the original article


[D. R. Stinson, G. H. J. van Rees, The equivalence of certain equidistant binary codes and symmetric BIBDs, *Combinatorica,* **4** (1984) 357-362](http://link.springer.com/article/10.1007/BF02579148)


and also another article


[J. H. van Lint, On equidistant binary codes of lengthn $=4k+1$ with distanced $=2k$, *Combinatorica,* **4** (1984) 321-323](http://link.springer.com/article/10.1007/BF02579143)) in the same issue.


There are more interesting direct connections between codes and designs when it comes to equidistant codes, and this is true for the non-binary case as well. If I remember correctly, the book chapter


V. D. Tonchev, `Codes and designs', in *Handbook of Coding Theory Vol II,* edited by V. S. Pless and W. C. Human (North Holland,
Amsterdam, 1998) Chap. 15, pp. 1229-1268


has a section for equidistant codes and their relations to combinatorial designs. Basically, the optimal $q$-ary equidistant codes are equivalent to resolvable designs in the language of design theory.


Now getting back to the binary case, if you're also interested in the case when there's a slight gap between $a$ and $b$, you can still employ design theory. In fact, such codes achieving the [Johson bound](http://en.wikipedia.org/wiki/Johnson_bound) can be constructed rather easily for $a+1 = b$ by using almost difference sets and whatnot straightforwardly. Basically, you try to construct a design that is nearly symmetric but not quite. Almost difference sets are quite important for synchronization and many other problems. You should be able to find many results if you look around. You can widen the gap a bit more, too. I'll explain how in the remainder of this post by giving a construction for the case when $a+2=b$ (Basically the same idea also works for $a+1=b$ etc.).


For instance, a *Steiner* $2$-*design* of *order* $v$ and *block size* $k$ is a pair $(V, \mathcal{B})$ of finite sets, where $\mathcal{B}$ is a set of $k$-subsets of $V$ of cardinality $\vert V \vert = v$ such that any distinct pair $a, b \in V$ is contained in exactly one element of $\mathcal{B}$. If you regard each element $B \in \mathcal{B}$ as the support of a codeword of a binary code of length $v$, you get a code in which any pair of distinct codewords is of distance $2k$ or $2(k-1)$. This code attains the Johnson bound with equality because a Steiner $2$-design packs all pairs into the elements of $\mathcal{B}$ without duplicates. Wilson's *Fundamental Existence Theory* asserts that you can construct Steiner $2$-designs for all sufficiently large orders (as long as certain necessary conditions by a counting argument are met). So it's a good source of optimal examples of $a+2=b$. This idea generally works for packings, Steiner $t$-designs, and the like, so you can get more examples or increase the gap between $a$ and $b$ more. Since there is a well-developed existence theory for these kinds of combinatorial designs, you can prove the existence of optimal examples and actually construct them.


By the way, the latest issue of the Journal of Combinatorial Theory A contains a paper on the asymptotic existence of $2$-designs, and this result can be directly used exactly the same way I explained above for your purpose: 


[Y. M. Chee, C. J. Colbourn, A. C. H. Ling, R. M. Wilson, Covering and packing for pairs, *J. Combin. Theory Ser. A* **120** (2013) 1440–1449](http://www.sciencedirect.com/science/article/pii/S0097316513000691)


Edit: Oops, I forgot to mention this, but when a binary code is *not* equidistant, you may not be able to turn it into a constant-weight one with one less codeword. So the latter half of this post is assuming that the actual bound is probably the same or not far away from the Johnson bound (which is a bound on the number of codewords of a constant-wight code) if $a$ and $b$ are pretty close. Also, I'm not familiar with the case when $b$ is large (compared to $a$). If you impose a condition such as the code must be linear and have a pair of codewords of distance $b = n$, then I think you can find results if you google "linear self-complementary codes."


