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Ahlswede and Katona (1977) posed the following isodiametric problem in Hamming spaces: For every $n$ and $1le Mle2^{n}$, 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-Yeungs bound by finding a better feasible solution to their dual program. For fixed $0<ale1/2$ and for $M=leftlceil a2^{n}rightrceil $, our feasible solution attains the asymptotically optimal value of Fu-Wei-Yeungs dual program as $ntoinfty$. Hence for $0<ale1/2$, all possible asymptotic bounds that can be derived by Fu-Wei-Yeungs 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.
$H_q(n,d)$ is defined as the graph with vertex set ${mathbb Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$
The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with sma
We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.
For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is a function $f : V to {0,1,2,3}$ having the property that if $f (v) = 0$, then vertex $v$ must have at least two neighbors assigned $2$ under $f$ or {at least} one neighbor $u$ with
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