A random construction for permutation codes and the covering radius

Keevash, Peter; Ku, Cheng Yeaw

doi:10.1007/s10623-006-0017-3

Cited by 21 publications

(37 citation statements)

References 13 publications

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“…Note that we are using a special and uncommon version of the lemma whose applications can be found in the proof on Latin transversals in [1], in [9], or in [2].…”

Section: Collection Of Events and For Anymentioning

confidence: 99%

“…Another dimension for further research stems from a certain semi-random construction that was considered by Ku-Keevash in [9]. In their paper, they utilized Theorem 3.4 to prove the existence of a randomized algorithm that, under certain conditions, constructs sets of permutations (from the underlying symmetric group S n ) that are <s-intersecting; that is, the sets constitute codes whose minimum distances are at least n − s + 1.…”

Section: Algorithmic and Asymptotic Concernsmentioning

confidence: 99%

“…In their paper, they utilized Theorem 3.4 to prove the existence of a randomized algorithm that, under certain conditions, constructs sets of permutations (from the underlying symmetric group S n ) that are <s-intersecting; that is, the sets constitute codes whose minimum distances are at least n − s + 1. Clearly, it is possible to consider this problem for our metric space (Ω, d) of 1-factors of the complete uniform hypergraph H ; in fact one may wish to consider analogues of the randomized algorithm in [9] for (Ω, d). More generally, we would like to see such problems considered for codes on any finite metric space.…”

Section: Algorithmic and Asymptotic Concernsmentioning

confidence: 99%

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The covering radius problem for sets of 1-factors of the complete uniform hypergraphs

2015

Discrete Mathematics

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a b s t r a c tCovering and packing problems constitute a class of questions concerning finite metric spaces that have surfaced in recent literature. In this paper, we consider, for the first time, these problems for the finite metric space (Ω, d) arising from the set Ω of 1-factors of the complete t-uniform hypergraph H on nt vertices for some positive integers n and t. We focus on the covering problem; in particular we investigate bounds on the covering radius of any code C ⊆ Ω. In doing so, we give both upper and lower bounds on the covering radius, as well as a frequency parameter type result that follows from the Lovász local lemma.

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“…Note that we are using a special and uncommon version of the lemma whose applications can be found in the proof on Latin transversals in [1], in [9], or in [2].…”

Section: Collection Of Events and For Anymentioning

confidence: 99%

Section: Algorithmic and Asymptotic Concernsmentioning

confidence: 99%

Section: Algorithmic and Asymptotic Concernsmentioning

confidence: 99%

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The covering radius problem for sets of 1-factors of the complete uniform hypergraphs

2015

Discrete Mathematics

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“…An important result for PAs with minimum distances that are far away from both n and 0 appears in Keevash and Ku (2006). There, it is shown that a randomised algorithm can be used to produce a set of permutations subject to certain conditions.…”

Section: Introductionmentioning

confidence: 99%

Classes of permutation arrays in finite projective spaces

Alderson

Mellinger

2010

IJICOT

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We exhibit some techniques for constructing permutation arrays using projections in finite projective spaces and the geometry of arcs in the finite projective plane. We say a permutation array PA(n, d) has length n and minimum distance d when it consists of a collection of permutations on n symbols that pairwise agree in at most n − d coordinate positions. Such arrays can also be viewed as non-linear codes and are used in powerline communication. While our techniques likely do not produce optimal arrays, we are able to construct examples of codes for certain parameter sets for which no constructions were previously known.

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“…We refer the reader to [2,6] for related algebraic results, to [10] for a nice probabilistic approach, and to [3,15] for some combinatorial bounds.…”

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confidence: 99%

Bounds on permutation codes of distance four

Dukes

Sawchuck

2009

J Algebr Comb

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A permutation code of length n and distance d is a set of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ is at least d. In this note, we determine some new results on the maximum size of a permutation code with distance equal to 4, the smallest interesting value. The upper bound is improved for almost all n via an optimization problem on Young diagrams. A new recursive construction improves known lower bounds for small values of n.

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A random construction for permutation codes and the covering radius

Abstract: We analyse a probabilistic argument that gives a semi-random construction for a permutation code on n symbols with distance n − s and size (s!(log n) 1/2 ), and a bound on the covering radius for sets of permutations in terms of a certain frequency parameter.

Cited by 21 publications

References 13 publications

The covering radius problem for sets of 1-factors of the complete uniform hypergraphs

The covering radius problem for sets of 1-factors of the complete uniform hypergraphs

Classes of permutation arrays in finite projective spaces

Bounds on permutation codes of distance four

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