Construction of (n,r)-arcs in PG(2,q)

Braun, Michael; Kohnert, Axel; Wassermann, Alfred

doi:10.2140/iig.2005.1.133

Cited by 29 publications

(45 citation statements)

References 10 publications

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“…This construction method is a general approach that works for many discrete structures as designs [14,3], q-analogs of designs [6,7], arcs in projective geometries [8], linear codes [2,4,5,15] or quantum codes [21].…”

Section: Constant Dimension Codes With Prescribed Automorphismsmentioning

confidence: 99%

Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance

Kohnert

Kurz

2008

Mathematical Methods in Computer Science

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127

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In this paper we construct constant dimension space codes with prescribed minimum distance. There is an increased interest in space codes since a paper [13] by Kötter and Kschischang were they gave an application in network coding. There is also a connection to the theory of designs over finite fields. We will modify a method of Braun, Kerber and Laue [7] which they used for the construction of designs over finite fields to do the construction of space codes. Using this approach we found many new constant dimension spaces codes with a larger number of codewords than previously known codes. We will finally give a table of the best found constant dimension space codes. network coding, q-analogue of Steiner systems, subspace codes

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Section: Constant Dimension Codes With Prescribed Automorphismsmentioning

confidence: 99%

Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance

Kohnert

Kurz

2008

Mathematical Methods in Computer Science

Self Cite

127

160

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“…This construction method is a general approach that works for many discrete structures as designs [15,2], q-analogs of designs [6], arcs in projective geometries [7] or linear codes [1,4,5]. The general method is as follows: The matrix M is reduced by adding up columns (labeled by the points of P HG(2, GR(p s , p sm )) corresponding to the orbits of G. Now because of the relation…”

Section: Lemma 2 (Recursive Construction)mentioning

confidence: 99%

Sets of Type (d 1,d 2) in Projective Hjelmslev Planes over Galois Rings

Kohnert

2009

Algorithmic Algebraic Combinatorics and Gröbner Bases

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Summary. In this paper we construct sets of type (d1, d2) in the projective Hjelmslev plane. For computational purposes we restrict ourself to planes over Zps with p a prime and s > 1, but the method is described over general Galois rings. The existence of sets of type (d1, d2) is equivalent to the existence of a solution of a Diophantine system of linear equations. To construct these sets we prescribe automorphisms, which allows to reduce the Diophantine system to a feasible size. At least two of the newly constructed sets are 'good' u−arcs. The size of one of them is close to the known upper bound.

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“…These codes have a wide range of applications, and they tie in well with structures in projective geometry. The compact disc stores music using linear (32,28,5) and (28,24,5)-MDS-codes over F 2 8 (for details see Chapter 5). Trivial MDS-codes are the codes of types (n, 1, n), (n, n − 1, 2), and (n, n, 1), which exist over any field F q (cf.…”

Section: Mds-codesmentioning

confidence: 99%

“…At this point, we only mention that two codes which are defined over F 2 8 play an important role. These codes, with parameters (32, 28,5) and (28,24,5) are obtained from a (255, 251, 5)-Reed-Solomon-code over F 2 8 by successive shortening. The encoding with respect to these two codes is completely explained in Section 5.4.…”

Section: Examplesmentioning

confidence: 99%

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Error-Correcting Linear Codes

Betten¹,

Braun²,

Fripertinger³

et al. 2006

Algorithms and Computation in Mathematics

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Preface IXcodes. The largest possible minimum distance is given, together with the number of semilinear isometry classes of such optimal codes. In addition, corresponding generator matrices can be found. Altogether, around 2 million isometry classes of codes have been computed, of which more than 800 000 are optimal codes. Nearly 200 000 generator matrices can be found on the attached compact disc, of which more than 70 000 generate optimal codes. The complete set of computed generator matrices can be downloaded from the web-site mentioned above. These codes are all pairwise inequivalent. More precisely, they are representatives of different semilinear isometry classes.On the side of the reader we assume only a basic knowledge of Linear Algebra and Algebra. Many fundamental notions are reviewed in the text. Readers with a background in field theory may skip Chapter 3. We should also mention what this book for one reason or the other does not cover. For instance, we do not discuss algebraic geometric codes, in particular the generalized version of Goppa-codes. Also not included are convolutional codes, Turbo codes, LDPC codes, codes over rings, e.g. Z 4 , and decoding methods using Tannergraphs. All this, as well as the connection between codes and designs and a deeper account on the theory of self-dual codes had to be left out. To this end, we refer the interested reader to the excellent literature on these topics, for instance the recent books by Pless and Huffman [94], Moon [153], Nebe, Rains and Sloane [157]. The "classic" for nearly 30 years, the book by MacWilliams and Sloane [139] from 1977 is still astonishingly comprehensive. The connections between codes and designs are described in the book by Assmus and Key [5]. The Handbook of Coding Theory [163], edited by Pless and Huffman, has articles on many of these topics written by experts in the field.The authors wish to express their sincerest thanks in particular to Karl-Heinz Zimmermann, coauthor of the German precursor book "Codierungstheorie". We also thank our colleague Reinhard Laue, to whom we owe great thanks for many stimulating discussions on the constructive theory of finite structures using group actions. Thanks are also due, of course, to our students and to the scientific community, in particular to Andries Brouwer, who maintains a very helpful table on parameters of optimal linear codes, as well as to the editors of the important two volumes of the Handbook of Coding Theory, Vera S. Pless and W. Cary Huffman. Furthermore, we greatly appreciate receiving helpful comments and suggestions from We acknowledge financial support by the Deutsche Forschungsgemeinschaft and theÖsterreichischer Fonds zur Förderung der wissenschaftlichen Forschung for X P r e f a c e helpful financial support of several research projects on these topics. These projects contributed very much to the development of the theory and to the implementation of corresponding software, as well as to the collection of data which are now available for the interested reader via email,...

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Construction of (n,r)-arcs in PG(2,q)

Cited by 29 publications

References 10 publications

Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance

Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance

Sets of Type (d 1,d 2) in Projective Hjelmslev Planes over Galois Rings

Error-Correcting Linear Codes

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