  Let $Q_{n}^{r}$ be the graph with vertex set $\{-1,1\}^{n}$ in which two
vertices are joined if their Hamming distance is at most $r$. The
edge-isoperimetric problem for $Q_{n}^{r}$ is that: For every $(n,r,M)$ such
that $1\le r\le n$ and $1\le M\le2^{n}$, determine the minimum edge-boundary
size of a subset of vertices of $Q_{n}^{r}$ with a given size $M$. In this
paper, we apply two different approaches to prove bounds for this problem. The
first approach is a linear programming approach and the second is a
probabilistic approach. Our bound derived by the first approach generalizes the
tight bound for $M=2^{n-1}$ derived by Kahn, Kalai, and Linial in 1989.
Moreover, our bound is also tight for $M=2^{n-2}$ and $r\le\frac{n}{2}-1$. Our
bounds derived by the second approach are expressed in terms of the \emph{noise
stability}, and they are shown to be asymptotically tight as $n\to\infty$ when
$r=2\lfloor\frac{\beta n}{2}\rfloor+1$ and $M=\lfloor\alpha2^{n}\rfloor$ for
fixed $\alpha,\beta\in(0,1)$, and is tight up to a factor $2$ when
$r=2\lfloor\frac{\beta n}{2}\rfloor$ and $M=\lfloor\alpha2^{n}\rfloor$. In
fact, the edge-isoperimetric problem is equivalent to a ball-noise stability
problem which is a variant of the traditional (i.i.d.-) noise stability
problem. Our results can be interpreted as bounds for the ball-noise stability
problem.
