Cycle indices of linear, affine, and projective groups

Fripertinger, Harald

doi:10.1016/s0024-3795(96)00530-7

Cited by 23 publications

(24 citation statements)

References 8 publications

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“…The paper [4] contains all that is needed to work out numerical examples for Theorem 5.1. That paper also remotely hints at Theorem 5.1 via Lemma 5.2.…”

Section: Linear Codesmentioning

confidence: 99%

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Combinatorial families enumerated by quasi-polynomials

Lisonĕk

2007

Journal of Combinatorial Theory, Series A

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We say that the sequence (a n ) is quasi-polynomial in n if there exist polynomials P 0 , . . . , P s−1 and an integer n 0 such that, for all n n 0 , a n = P i (n) where i ≡ n (mod s). We present several families of combinatorial objects with the following properties: Each family of objects depends on two or more parameters, and the number of isomorphism types of objects is quasi-polynomial in one of the parameters whenever the values of the remaining parameters are fixed to arbitrary constants. For each family we are able to translate the problem of counting isomorphism types of objects into the problem of counting integer points in a union of parametrized rational polytopes. The families of objects to which this approach is applicable include combinatorial designs, linear and unrestricted codes, and dissections of regular polygons.

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“…The paper [4] contains all that is needed to work out numerical examples for Theorem 5.1. That paper also remotely hints at Theorem 5.1 via Lemma 5.2.…”

Section: Linear Codesmentioning

confidence: 99%

“…Formulas for the cycle indices of linear groups were indeed computed in [4] and they are implemented in the software system SYMMETRICA developed at the University of Bayreuth. It is thus a routine matter to find, for example, that the o.g.f.…”

Section: Linear Codesmentioning

confidence: 99%

Combinatorial families enumerated by quasi-polynomials

Lisonĕk

2007

Journal of Combinatorial Theory, Series A

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“…These cycle indices are known for q = 2, see [50], [60], [82], [83], [184], and programs for their evaluation are implemented in SYMMETRICA ( [190]), so that tables can be determined easily. Comparing Tables 6.2 and 6.1 shows that the set of isometry classes of (n, k)-codes is much smaller than the set of of all (n, k)-codes for given parameters n and k. Q If the cycle indices C(PGL k (q), PG * k−1 (q)) are known for general q, it is possible to evaluate the numbers T nkq and T nkq , from which we can deduce V nkq , V nkq , U nkq , and U nkq for arbitrary fields F q .…”

Section: 124mentioning

confidence: 99%

“…. , f t−1 are pairwise distinct monic, irreducible polynomials over F. If there exists a decomposition 6.3.9 of F k with exactly a A different approach to normal forms can be found in [60].…”

Section: 37mentioning

confidence: 99%

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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“…, P n } such that P i ∈ P N (k) for all i. This is due to the fact that these orbits are in bijective correspondence with isometry classes of certain linear codes [2,5,7,9,10]. However, to our knowledge, the enumeration of rational n-sets has been considered so far only in dimension 1; for instance in [8], where the numbers t 1 (n) were computed.…”

Section: Introductionmentioning

confidence: 99%