  Ahlswede and Katona (1977) posed the following isodiametric problem in
Hamming spaces: For every $n$ and $1\le M\le2^{n}$, determine the minimum
average Hamming distance of binary codes with length $n$ and size $M$. Fu, Wei,
and Yeung (2001) used linear programming duality to derive a lower bound on the
minimum average distance. However, their linear programming approach was not
completely exploited. In this paper, we improve Fu-Wei-Yeung's bound by finding
a better feasible solution to their dual program. For fixed $0<a\le1/2$ and for
$M=\left\lceil a2^{n}\right\rceil $, our feasible solution attains the
asymptotically optimal value of Fu-Wei-Yeung's dual program as $n\to\infty$.
Hence for $0<a\le1/2$, all possible asymptotic bounds that can be derived by
Fu-Wei-Yeung's linear program have been characterized. Furthermore, noting that
the average distance of a code is closely related to weights of Fourier
coefficients of a Boolean function, we also apply the linear programming
technique to prove bounds on Fourier weights of a Boolean function of various
degrees.
