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Coding schemes for crisscross error patterns

WIRELESS PERSONAL COMMUNICATIONS 2008年第1期47卷 39-49页

作者： Plass, Simon Richter, Gerd Vinck, A. J. Han German Aerosp Ctr DLR Inst Commun & Nav D-82234 Wessling Germany Univ Ulm Dept TAIT D-89081 Ulm Germany Univ Essen Gesamthsch Inst Expt Math D-45326 Essen Germany

This paper addresses two coding schemes which can handle emerging errors with crisscross patterns. First, a code with maximum rank distance, so-called Rank-codes, is described and a modified Berlekamp-Massey algorithm is provided. Secondly, a permutation code based coding scheme for crisscross error patterns is presented. The influence of different types of noise are also discussed.

关键词： Rank codes permutation code crisscross errors

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Composition Check codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第1期64卷 249-256页

作者： Immink, Kees A. Schouhamer Cai, Kui Turing Machines Inc NL-3016 DK Rotterdam Netherlands Singapore Univ Technol & Design Singapore 487372 Singapore

We present composition check codes for noisy storage and transmission channels with unknown gain and/or offset. In the proposed composition check code, like in systematic error correcting codes, the encoding of the main data into a constant composition code is completely avoided. To the main data, a coded label is appended that carries information regarding the composition vector of the main data. Slepian's optimal detection technique of codewords that are taken from a constant composition code is applied for detection. A first Slepian detector detects the label and subsequently restores the composition vector of the main data. The composition vector, in turn, is used by a second Slepian detector to optimally detect the main data. We compute the redundancy and error performance of the new method, and results of computer simulations are presented.

关键词： Constant composition code permutation code flash memory optical recording

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A Novel Representation for permutations

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IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第3期67卷 1920-1927页

作者： Ri, Won-Ho Song, Ok-Hyon Kim Il Sung Univ Fac Math Pyongyang North Korea

In this paper, we present a variable-length binary code for permutations of degree n, where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how "well shuffled" a permutation is. permutations with longer codewords seem more "random." An enumeration method of the variable-length code is also given by using T. Cover's enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary n in a straightforward way.

关键词： Variable-length code permutation code enumerative coding source coding

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Coding with injections

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DESIGNS codeS AND CRYPTOGRAPHY 2012年第3期65卷 213-222页

作者： Dukes, Peter J. Univ Victoria Victoria BC Canada

A permutation code of length n and minimum distance d is a set Gamma of permutations from some fixed set of n symbols such that the Hamming distance between any distinct u, upsilon is an element of Gamma is at least d. As a generalization, we introduce the problem of packing injections from an m-set, m <= n, sometimes called m-arrangements, relative to Hamming distance. We offer some preliminary coding-theoretic bounds, a few design-theoretic connections, and a short discussion on possible applications.

关键词： permutation code Injection Arrangement Ordered design Hamming distance

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On constant composition codes

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DISCRETE APPLIED MATHEMATICS 2006年第6期154卷 912-929页

作者： Chu, WS Colbourn, CJ Dukes, P Arizona State Univ Dept Comp Sci & Engn Tempe AZ 85287 USA Univ Victoria Dept Math & Stat Victoria BC V8W 3P4 Canada

A constant composition code over a k-ary alphabet has the property that the numbers of occurrences of the k symbols within a codeword is the same for each codeword. These specialize to constant weight codes in the binary case, and permutation codes in the case that each symbol occurs exactly once. Constant composition codes arise in powerline communication and balanced scheduling, and are used in the construction of permutation codes. In this paper, direct and recursive methods are developed for the construction of constant composition codes. (c) 2005 Elsevier B.V. All rights reserved.

关键词： constant composition code constant weight code permutation code

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On the Algebraic Combinatorics of Injections and its Applications to Injection codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第11期66卷 6898-6907页

作者： Dukes, Peter J. Ihringer, Ferdinand Lindzey, Nathan Univ Victoria Dept Math & Stat Victoria BC V8P 5C2 Canada Univ Ghent Dept Math Anal Log & Discrete Math B-9000 Ghent Belgium Univ Colorado Dept Comp Sci Boulder CO 80309 USA

We consider the algebraic combinatorics of the set of injections from a k-element set to an n-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair (S-k x S-n, diag(S-k) x Sn- k). We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.

关键词： Standards Shape Strips Hamming distance Linear programming Eigenvalues and eigenfunctions Indexes permutation code linear programming bound injection partial permutation representation theory association scheme

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Minimum Pearson Distance Detection for Multilevel Channels With Gain and/or Offset Mismatch

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IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第10期60卷 5966-5974页

作者： Immink, Kees A. Schouhamer Weber, Jos H. Turing Machines Inc NL-3016 DK Rotterdam Netherlands Delft Univ Technol Dept Elect Engn Math & Comp Sci NL-2628 CN Delft Netherlands

The performance of certain transmission and storage channels, such as optical data storage and nonvolatile memory (flash), is seriously hampered by the phenomena of unknown offset (drift) or gain. We will show that minimum Pearson distance (MPD) detection, unlike conventional minimum Euclidean distance detection, is immune to offset and/or gain mismatch. MPD detection is used in conjunction with T-constrained codes that consist of q-ary codewords, where in each codeword T reference symbols appear at least once. We will analyze the redundancy of the new q-ary coding technique and compute the error performance of MPD detection in the presence of additive noise. Implementation issues of MPD detection will be discussed, and results of simulations will be given.

关键词： Constant composition code permutation code rank modulation flash memory digital optical data storage recording non-volatile memory NVM mismatch adaptive equalisation fading Pearson distance Euclidean distance fading

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Some new resolvable GDDs with k=4 and doubly resolvable GDDs with k=3

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DISCRETE MATHEMATICS 2015年第11期338卷 2105-2118页

作者： Du, Juan Abel, R. Julian R. Wang, Jinhua Nantong Univ Sch Sci Nantong 226007 Peoples R China Univ New S Wales Sch Math & Stat Sydney NSW 2052 Australia

A doubly resolvable packing design with block size k, index lambda, replication number r, and nu elements is called a generalized Kirkman square and denoted by GKS(k) (nu;1, lambda;r). Existence of GKS(3) (4u;1, 1;2(u - 1))s and GKS(3)(6u;1, 1;3(u - 1))s is implied by existence of doubly resolvable group divisible designs with block size 3, index 1, and types 4(u) and 6(u) (i.e., (3, 1)-DRGDDs of types 4(u) and 6(u). In this paper, we establish the spectra of (3, 1)-DRGDDs of types 4(u) and 6(u) with 15 and 31 possible exceptions, respectively. As applications, we get some new classes of permutation codes and doubly constant weight codes. We also construct 5 new resolvable GDDs with block size 4 and index 1. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Group divisible design Resolvable Doubly resolvable Generalized Kirkman square Frame permutation code Doubly constant weight code

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Are random permutations spherically uniform?

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ELECTRONIC JOURNAL OF PROBABILITY 2020年第none期25卷 1-26页

作者： Perlman, Michael D. Univ Washington Dept Stat Seattle WA 98195 USA

For large q, does the (discrete) uniform distribution on the set of q! permutations of the vector (X) over bar (q) = (1, 2, ..., q)' closely approximate the (continuous) uniform distribution on the (q - 2)-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a (q - 1)-dimensional convex polyhedron. The answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, (1, 2, ..., q) is not the most favorable configuration for spherical uniformity of permutations. A more favorable configuration (X) over cap (q) is found, namely that which minimizes the normalized surface area of the largest empty spherical cap among its q! permutations. Unlike that for (X) over bar (q), the normalized surface area of the largest empty spherical cap among the permutations of (X) over cap (q) approaches 0 as q -> infinity. Nonetheless the permutations of (X) over cap (q) do not approach spherical uniformity either. The existence of an asymptotically spherically uniform permutation sequence remains an open question.

关键词： permutations uniform distribution spherical cap discrepancy largest empty cap regular configuration regular permutohedron L-minimal configuration L-minimal permutohedron normal configuration normal permutohedron majorization spherical code permutation code

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Lehmer code transforms and Mahonian statistics on permutations

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DISCRETE MATHEMATICS 2013年第5期313卷 581-589页

作者： Vajnovszki, Vincent Univ Bourgogne LE2I F-21078 Dijon France

In 2000 Babson and Steingrimsson introduced the notion of vincular patterns in permutations. They show that essentially all well-known Mahonian permutation statistics can be written as combinations of such patterns. Also, they proved and conjectured that other combinations of vincular patterns are still Mahonian. These conjectures were proved later: by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In this paper we give an alternative proof of some of these results. Our approach is based on permutation codes which, like the Lehmer code, map bijectively permutations onto subexcedant sequences. More precisely, we give several code transforms (i.e., bijections between subexcedant sequences) which when applied to the Lehmer code yield new permutation codes which count occurrences of some vincular patterns. These code transforms can be seen as a pre-compression step of the Lehmer code because they map some redundancies into runs of 0s. Also, our proofs, unlike the previous ones, provide explicit bijections between permutations having a given value for two different Mahonian pattern-based statistics. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Mahonian statistics permutation code (Combinations of) vincular patterns

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