An Improved Linear Programming Bound on the Average Distance of a Binary Code
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Ahlswede and Katona (1977) posed the following isodiametric problem in Hamming spaces: For every n and 1< M<2^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<1/2 and for M= a2^n, our feasible solution attains the asymptotically optimal value of Fu-Wei-Yeung's dual program as n→∞. Hence for 0<a<1/2, 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.
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