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

Yu, Lei; Tan, Vincent Y. F.

doi:10.48550/arxiv.1910.09416

Cited by 6 publications

(10 citation statements)

References 14 publications

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“…The bounds t 2 and 2t 2 1 √ t − 1 were respectively proven by Fu, Wei, and Yeung [31] and the present author and Tan [32], by using linear programming methods (together with MacWilliams-Delsarte identities). The bound 2t 2 ln 1 t is called Chang's bound, which was proven by using hypercontractivity inequalities [13].…”

Section: B Case Of α ∈ [2 3]mentioning

confidence: 68%

On the $Φ$-Stability and Related Conjectures

Lei¹

2021

Preprint

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Let X be a random variable uniformly distributed on the discrete cube {−1, 1}n , and let T ρ be the noise operator acting on Boolean functions f : {−1, 1} n → {0, 1}, where ρ ∈ [0, 1] is the noise parameter, representing the correlation coefficient between each coordination of X and its noise-corrupted version. Given a convex function Φ and the mean Ef (X) = a ∈ [0, 1], which Boolean function f maximizes the Φ-stability E [Φ (T ρ f (X))] of f ? Special cases of this problem include the (symmetric and asymmetric) α-stability problems and the "Most Informative Boolean Function" problem. In this paper, we provide several upper bounds for the maximal Φ-stability. Considering specific Φ's, we partially resolve Mossel and O'Donnell's conjecture on α-stability with α > 2, Li and Médard's conjecture on α-stability with 1 < α < 2, and Courtade and Kumar's conjecture on the "Most Informative Boolean Function" which corresponds to a conjecture on α-stability with α = 1. Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut-Kalai-Naor (FKN) theorem. Our improvements of the FKN theorem are sharp or asymptotically sharp for certain cases.

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Section: B Case Of α ∈ [2 3]mentioning

confidence: 68%

On the $Φ$-Stability and Related Conjectures

Lei¹

2021

Preprint

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“…Other lower bounds on dpN q are given in [16] and subsequent works, with the best known results appearing in the recent paper [31].…”

Section: 3mentioning

confidence: 98%

Stolarsky's invariance principle for finite metric spaces

Barg

2020

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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.

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“…, where the upper bound is attained by symmetric (n−2)-subcubes (e.g., A = B = {x : x 1 = x 2 = 1}). Next, the first author and Tan [22] strengthened the bounds in [2] by improving Fu-Wei-Yeung's linear programming bound on the average distance. Numerical results show that the upper bound in [22] is strictly tighter than existing bounds for the case P n X (A) = P n Y (B) = 1 8 .…”

Section: B Noninteractive Simulation With Sources P N Xymentioning

confidence: 99%

“…Next, the first author and Tan [22] strengthened the bounds in [2] by improving Fu-Wei-Yeung's linear programming bound on the average distance. Numerical results show that the upper bound in [22] is strictly tighter than existing bounds for the case P n X (A) = P n Y (B) = 1 8 . Kahn, Kalai, and Linial [23] first applied the single-function version of (forward) hypercontractivity inequalities to obtain bounds for the noninteractive simulation problem, by substituting the nonnegative functions in the hypercontractivity inequalities with the Boolean functions.…”

Section: B Noninteractive Simulation With Sources P N Xymentioning

confidence: 99%

Graphs of Joint Types, Noninteractive Simulation, and Stronger Hypercontractivity

Yu¹,

Anantharam²

2021

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In this paper, we introduce the concept of a type graph, namely a bipartite graph induced by a joint type. We study the maximum edge density of induced bipartite subgraphs of this graph having a number of vertices on each side on an exponential scale. This can be seen as an isoperimetric problem. We provide asymptotically sharp bounds for the exponent of the maximum edge density as the blocklength goes to infinity. We also study the biclique rate region of the type graph, which is defined as the set of (R 1 , R 2 ) such that there exists a biclique of the type graph which has respectively e nR1 and e nR2 vertices on two sides. We provide asymptotically sharp bounds for the biclique rate region as well. We then apply our results and proof ideas to noninteractive simulation problems. We completely characterize the exponents of maximum and minimum joint probabilities when the marginal probabilities vanish exponentially fast with given exponents. These results can be seen as strong small-set expansion theorems. We extend the noninteractive simulation problem by replacing Boolean functions with arbitrary nonnegative functions, and obtain new hypercontractivity inequalities which are stronger than the common hypercontractivity inequalities. Furthermore, as an application of our results, a new outer bound for the zero-error capacity region of the binary adder channel is provided, which improves the previously best known bound, due to Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on the method of types, linear algebra, and coupling techniques.

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

Cited by 6 publications

References 14 publications

On the $Φ$-Stability and Related Conjectures

On the $Φ$-Stability and Related Conjectures

Stolarsky's invariance principle for finite metric spaces

Graphs of Joint Types, Noninteractive Simulation, and Stronger Hypercontractivity

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