A Survey on Packing and Covering Problems in the Hamming Permutation Space

Quistorff, Jörn

doi:10.37236/1161

Cited by 22 publications

(20 citation statements)

References 26 publications

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“…The most traditional approach in the first category is to place as many balls of radius d/2 into S n as possible. Similarly one can do a clique search in a graph G(n, d) whose vertices are elements in S n and edges are connected between two permutations with Hamming distance ≥ d. A greedy algorithm and a search via automorphisms can be also applied as shown in [6] or [18]. These constructions are based basically on an exhaustive search; hence it is hard to find the row for any given index unless we store the whole array.…”

Section: Previous Resultsmentioning

confidence: 99%

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A New Construction of Permutation Arrays

Park¹,

Song

2012

IEICE Trans. Fundamentals

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Let PA(n, d) be a permutation array (PA) of order n and the minimum distance d. We propose a new construction of the permutation array PA p m , p m−1 k for a given prime number p, a positive integer k < p and a positive integer m. The resulted array has |PA(p, k)| • p (m−1)(p−k) m rows. Compared to the other constructions, the new construction gives a permutation array of far bigger size with a large minimum distance, for example, when k ≥ 2p/3. Moreover the proposed construction provides an algorithm to find the i-th row of PA p m , p m−1 k for a given index i very simply.

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Section: Previous Resultsmentioning

confidence: 99%

“…In the research of permutation arrays, the most important issue is to construct a permutation array with a large number of rows for given n and d, see e.g., [1], [2], [4], [6]- [12] and [18]. Another issue, which is considered less in literature, is to find the row of the array corresponding to a given index.…”

Section: Introductionmentioning

confidence: 99%

A New Construction of Permutation Arrays

Park¹,

Song

2012

IEICE Trans. Fundamentals

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“…The embeddability of a permutation code of size n 2 − n − 1 (case δ = 1 of Theorem 6) had been shown by Quistorff [13]. Of particular interest is case n = 10, the smallest integer n > 6 for which no projective plane of order n exists.…”

Section: Definitionmentioning

confidence: 98%

A Bound on Permutation Codes

Bierbrauer

Metsch

2013

Electron. J. Combin.

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Consider the symmetric group $S_n$ with the Hamming metric. A permutation code on $n$ symbols is a subset $C\subseteq S_n.$ If $C$ has minimum distance $\geq n-1,$ then $\vert C\vert\leq n^2-n.$ Equality can be reached if and only if a projective plane of order $n$ exists. Call $C$ embeddable if it is contained in a permutation code of minimum distance $n-1$ and cardinality $n^2-n.$ Let $\delta =\delta (C)=n^2-n-\vert C\vert$ be the deficiency of the permutation code $C\subseteq S_n$ of minimum distance $\geq n-1.$We prove that $C$ is embeddable if either $\delta\leq 2$ or if $(\delta^2-1)(\delta +1)^2<27(n+2)/16.$ The main part of the proof is an adaptation of the method used to obtain the famous Bruck completion theorem for mutually orthogonal latin squares.

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“…Theorem 23. f (n, 2) ≤ 4 3 n + O(1) for all n. The reader is encouraged to seek out [8] and the survey by Quistorff [44] for more information on covering radii for sets of permutations.…”

Section: Covering Radii For Sets Of Permutationsmentioning

confidence: 99%

Transversals in latin squares: a survey

Wanless¹

2011

Surveys in Combinatorics 2011

123

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A latin square of order n is an n × n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutually orthogonal latin squares.Here we provide a brief survey of the literature on transversals. We cover (1) existence and enumeration results, (2) generalisations of transversals including partial transversals and plexes, (3) the special case when the latin square is a group table, (4) a connection with covering radii of sets of permutations. The survey includes a number of conjectures and open problems.2000 Mathematics Subject Classifications: 05B15 20N05

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A Survey on Packing and Covering Problems in the Hamming Permutation Space

Abstract: Consider the symmetric group $S_n$ equipped with the Hamming metric $d_H$. Packing and covering problems in the finite metric space $(S_n,d_H)$ are surveyed, including a combination of both.

Cited by 22 publications

References 26 publications

A New Construction of Permutation Arrays

A New Construction of Permutation Arrays

A Bound on Permutation Codes

Transversals in latin squares: a survey

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