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Designs, Codes and Cryptography - Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial... 相似文献
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For z1,z2,z3∈Zn, the tristance d3(z1,z2,z3) is a generalization of the L1-distance on Zn to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticodeAd of diameter d is a subset of Zn with the property that d3(z1,z2,z3)?d for all z1,z2,z3∈Ad. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in Z2 for all diameters d?1. We then generalize this result to two related distance models: a different distance structure on Z2 where d(z1,z2)=1 if z1,z2 are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z2 is replaced by the hexagonal lattice A2. We also investigate optimal tristance anticodes in Z3 and optimal quadristance anticodes in Z2, and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go. 相似文献
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Tuvi Etzion 《组合设计杂志》2008,16(2):137-151
A doubly constant weight code is a binary code of length n1 + n2, with constant weight w1 + w2, such that the weight of a codeword in the first n1 coordinates is w1. Such codes have applications in obtaining bounds on the sizes of constant weight codes with given minimum distance. Lower and upper bounds on the sizes of such codes are derived. In particular, we show tight connections between optimal codes and some known designs such as Howell designs, Kirkman squares, orthogonal arrays, Steiner systems, and large sets of Steiner systems. These optimal codes are natural generalization of Steiner systems and they are also called doubly Steiner systems. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 137–151, 2008 相似文献
4.
Given an (n, k) linear code
over GF(q), the intersection of
with a codeπ(
), whereπSn, is an (n, k1) code, where max{0, 2k−n}k1k. The intersection problem is to determine which integers in this range are attainable for a given code
. We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we give a complete solution to the intersection problem for most of the interesting linear codes, e.g. cyclic codes, Reed–Muller codes, and most MDS codes. 相似文献
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Tuvi Etzion 《组合设计杂志》2007,15(1):15-34
The main goal of this article is to present several connections between perfect codes in the Johnson scheme and designs, and provide new tools for proving Delsarte conjecture that there are no nontrivial perfect Codes in the Johnson scheme. Three topics will be considered. The first is the configuration distribution which is akin to the weight distribution in the Hamming scheme. We prove that if there exists an e‐perfect code in the Johnson scheme then there is a formula which connects the number of vectors at distance i from any codeword in various codes isomorphic to . The second topic is the Steiner systems embedded in a perfect code. We prove a lower bound on the number of Steiner systems embedded in a perfect code. The last topic is the strength of a perfect code. We show two new methods for computing the strength of a perfect code and demonstrate them on 1‐perfect codes. We further discuss how to settle Delsarte conjecture. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 15–34, 2007 相似文献
6.
Motivated by applications in universal data compression algorithms we study the problem of bounds on the sizes of constant weight covering codes. We are concerned with the minimal sizes of codes of lengthn and constant weightu such that every word of lengthn and weightv is within Hamming distanced from a codeword. In addition to a brief summary of part of the relevant literature, we also give new results on upper and lower bounds to these sizes. We pay particular attention to the asymptotic covering density of these codes. We include tables of the bounds on the sizes of these codes both for small values ofn and for the asymptotic behavior. A comparison with techniques for attaining bounds for packing codes is also given. Some new combinatorial questions are also arising from the techniques.Part of this work was done while the first and third authors were visiting Bellcore. The third author was supported in part by National Science Foundation under grant NCR-8905052. Part of this work was presented in the Coding and Quantization Workshop at Rutgers University, NJ, October 1992. 相似文献
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Tuvi Etzion 《Designs, Codes and Cryptography》2014,72(2):405-421
Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph \(\mathcal{G }_q(n,r)\) by subspaces from the Grassmann graph \(\mathcal{G }_q(n,k)\) , \(k \ge r\) , are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, \(q\) -analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for \(q=2\) with \(r=2\) or \(r=3\) . We discuss the density for some of these coverings. Tables for the best known coverings, for \(q=2\) and \(5 \le n \le 10\) , are presented. We present some questions concerning possible constructions of new coverings of smaller size. 相似文献
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Tuvi Etzion 《Journal of Combinatorial Theory, Series A》1985,39(2):241-253
The distribution γ(c, n) of de Bruijn sequences of order n and linear complexity c is investigated. Some new results are proved on the distribution of de Bruijn sequences of low complexity, i.e., their complexity is between 2n?1 + n and 2n?1 + 2n?2. It is proved that for n ? 5 and 2n?1 + n?c<2n?1 + 2n?2, γ(c, n) ≡ 0 (mod 4). It is shown that for n ? 11, γ(2n?1 + n, n) > 0. It is also proved that γ(2n?1 + 2n?2, n) ? 4γ(2n?2 ? 1, n ? 2) and we give a recursive method to generate de Bruijn sequences of complexity 2n?1 + 2n?2. 相似文献
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Tuvi Etzion 《组合设计杂志》1994,2(5):359-374
Large sets of packings were investigated extensively. Much less is known about the dual problem, i.e., large sets of coverings. We examine two types of important questions in this context; what is the maximum number of disjoint optimal coverings? and what is the minimum number of optimal coverings for which the union covers the space? We give various constructions which give the optimal solutions and some good upper and lower bounds on both questions, respectively. © 1994 John Wiley & Sons, Inc. 相似文献
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T. Tuvi? I. Pa?ti S. Mentus 《Russian Journal of Physical Chemistry A, Focus on Chemistry》2011,85(13):2399-2405
Oxygen reduction reaction (ORR) was investigated in alkaline solution on tungsten electrode subjected to a previous anodic
dissolution. The rotating disk cyclic voltammetry and rotating disc chronoamperometry were used. Both unsupported and potassium
pechlorate and sulfate supported solutions were examined. The most striking feature of recorded ORR curves is the large difference
of ORR overpotential during anodic and cathodic sweep. This was attributed to the formation of tungsten oxide on the surface.
It was demonstrated that electrode pretreatment as well as the electrolyte composition greatly affects ORR electrochemistry
on tungsten electrode, and the influence of sulfates is discussed. 相似文献