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

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.Comment: 17 pages, 2 figure

Edge-Isoperimetric Inequalities and Ball-Noise Stability: Linear Programming and Probabilistic Approaches

Let $Q_{n}^{r}$ be the $r$-power of the hypercube $\{-1,1\}^{n}$. The discrete edge-isoperimetric problem for $Q_{n}^{r}$ is that: For every $\left(n,r,M\right)$ such that $1\le r\le n$ and $1\le M\le2^{n}$, determine the minimum 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. Our first approach is a linear programming approach and the second is a probabilistic approach. Our bound derived by using the first approach generalizes the sharp bound for $M=2^{n-1}$ derived by Kahn, Kalai, and Linial in 1989. Moreover, our bound is also sharp for $M=2^{n-2}$ and $r\le\frac{n-1}{2}$. Our bound derived by using the second approach is asymptotically sharp as $n\to\infty$ when $r=2\left\lfloor \frac{\beta n}{2}\right\rfloor +1$ and $M=\left\lfloor \alpha2^{n}\right\rfloor$ for fixed $\alpha,\beta\in(0,1)$, and sharp up to a constant factor when $r=2\left\lfloor \frac{\beta n}{2}\right\rfloor$ and $M=\left\lfloor \alpha2^{n}\right\rfloor$. Furthermore, the discrete 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 imply bounds on the ball-noise stability problem.Comment: 23 pages, no figure

Stolarsky's invariance principle for finite metric spaces

Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general structure behind various forms of the main identity. In this work we consider the case of finite metric spaces, relating the quadratic discrepancy of a subset to a certain function of the distribution of distances in it. Our main results are related to a concrete form of the invariance principle for the Hamming space. We derive several equivalent versions of the expression for the discrepancy of a code, including expansions of the discrepancy and associated kernels in the Krawtchouk basis. Codes that have the smallest possible quadratic discrepancy among all subsets of the same cardinality can be naturally viewed as energy minimizing subsets in the space. Using linear programming, we find several bounds on the minimal discrepancy and give examples of minimizing configurations. In particular, we show that all binary perfect codes have the smallest possible discrepancy.Comment: 24pp. v.2: Minor errors and typos corrected; v3: New closed formulas for the Krawtchouk expansion of lambda(w), new examples of discrepancy minimizers, added reference

On an inverse problem of the Erd\H{o}s-Ko-Rado type theorems

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is to characterize its extremal structure and its corresponding stability. The famous Erd\H{o}s-Ko-Rado theorem answered both direct and inverse problems and led the era of studying intersection problems for finite sets. In this paper, we consider the following quantitative intersection problem which can be viewed an inverse problem for Erd\H{o}s-Ko-Rado type theorems: For $\mathcal{F}\subseteq {[n]\choose k}$, define its \emph{total intersection} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in \mathcal{F}}|F_1\cap F_2|$. Then, what is the structure of $\mathcal{F}$ when it has the maximal total intersection among all families in ${[n]\choose k}$ with the same family size? Using a pure combinatorial approach, we provide two structural characterizations of the optimal family of given size that maximizes the total intersection. As a consequence, for $n$ large enough and $\mathcal{F}$ of proper size, these characterizations show that the optimal family $\mathcal{F}$ is indeed $t$-intersecting ($t\geq 1$). To a certain extent, this reveals the relationship between properties of being intersecting and maximizing the total intersection. Also, we provide an upper bound on $\mathcal{I}(\mathcal{F})$ for several ranges of $|\mathcal{F}|$ and determine the unique optimal structure for families with sizes of certain values.Comment: 36 page