Improved Lower Bounds on the Size of Balls over Permutations with the Infinity Metric

Moshe Schwartz; Pascal O. Vontobel

doi:https://doi.org/10.1109/TIT.2017.2697423

Improved Lower Bounds on the Size of Balls over Permutations with the Infinity Metric

Moshe Schwartz, Pascal O. Vontobel

Department of Electrical & Computer Engineering

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

We study the size (or volume) of balls in the metric space of permutations, Sn, under the infinity metric. We focus on the regime of balls with radius r = ρ · (n-1), ρ ϵ [0,1], i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0.029 bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.

Original language	American English
Article number	7908949
Pages (from-to)	6227-6239
Number of pages	13
Journal	IEEE Transactions on Information Theory
Volume	63
Issue number	10
DOIs	https://doi.org/10.1109/TIT.2017.2697423
State	Published - 1 Oct 2017

Keywords

Asymptotic gap
Sinkhorn theorem
infinity metric
permanent
permutation
rank modulation

All Science Journal Classification (ASJC) codes

Information Systems
Computer Science Applications
Library and Information Sciences

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Cite this

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abstract = "We study the size (or volume) of balls in the metric space of permutations, Sn, under the infinity metric. We focus on the regime of balls with radius r = ρ · (n-1), ρ ϵ [0,1], i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0.029 bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.",

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AU - Vontobel, Pascal O.

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N2 - We study the size (or volume) of balls in the metric space of permutations, Sn, under the infinity metric. We focus on the regime of balls with radius r = ρ · (n-1), ρ ϵ [0,1], i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0.029 bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.

AB - We study the size (or volume) of balls in the metric space of permutations, Sn, under the infinity metric. We focus on the regime of balls with radius r = ρ · (n-1), ρ ϵ [0,1], i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0.029 bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.

KW - Asymptotic gap

KW - Sinkhorn theorem

KW - infinity metric

KW - permanent

KW - permutation

KW - rank modulation

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JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

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M1 - 7908949

ER -

Improved Lower Bounds on the Size of Balls over Permutations with the Infinity Metric

Abstract

Keywords

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