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Ahlswede and Katona (1977) posed the following isodiametric problem inHamming spaces: For every $n$ and $1\le M\le2^{n}$, determine the minimumaverage 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 theminimum average distance. However, their linear programming approach was notcompletely exploited. In this paper, we improve Fu-Wei-Yeung's bound by findinga 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 theasymptotically 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 byFu-Wei-Yeung's linear program have been characterized. Furthermore, noting thatthe average distance of a code is closely related to weights of Fouriercoefficients of a Boolean function, we also apply the linear programmingtechnique to prove bounds on Fourier weights of a Boolean function of variousdegrees.

An Improved Linear Programming Bound on the Average Distance of a Binary Code