• Title, Summary, Keyword: double cosets

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INFINITE FAMILIES OF RECURSIVE FORMULAS GENERATING POWER MOMENTS OF TERNARY KLOOSTERMAN SUMS WITH SQUARE ARGUMENTS ASSOCIATED WITH O-(2n, q)

  • Kim, Dae-San
    • Journal of the Korean Mathematical Society
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    • v.48 no.2
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    • pp.267-288
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    • 2011
  • In this paper, we construct eight infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^-$(2n, q). Here q is a power of three. Then we obtain four infinite families of recursive formulas for power moments of Kloosterman sums with square arguments and four infinite families of recursive formulas for even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups $O^-$(2n, q).

CONSTRUCTION OF RECURSIVE FORMULAS GENERATING POWER MOMENTS OF KLOOSTERMAN SUMS: O+(2n, 2r) CASE

  • Kim, Dae San
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.585-602
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    • 2020
  • In this paper, we construct four infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the orthogonal group O+(2n, 2r). And we obtain two infinite families of recursive formulas for the power moments of Kloosterman sums and those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless' power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups O+(2n, 2r).

CANONICAL LEFT CELLS AND THE SHORTEST LENGTH ELEMENTS IN THE DOUBLE COSETS OF WEYL GROUPS

  • Kwon, Nam-Hee
    • Honam Mathematical Journal
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    • v.33 no.1
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    • pp.19-25
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    • 2011
  • Let G be the general linear group GL(n,$\mathbb{C}$), $W_0$ the Weyl group of G and W the extended a neWeyl group of G. Then it is well-known that W is a union of the double cosets $W_{0x}W_0$ as x moves over the set of dominant weights of W. It is also known that each double coset $W_{0x}W_0$ contains a unique element $m_x$ of the shortest length. These shortest length elements belong to what are called the canonical left cells. However, it is still an open problem to find the canonical left cell containing a given $m_x$. One of the mai purposes of this paper is to introduce a new approach to attack this question. In particular, we will present a conjecture which explicitly describes the canonical left cells containing an element $m_x$. We will show that our conjecture is true for some specific types of $m_x$.

ISOMORPHISM CLASSES OF CAYLEY PERMUTATION GRAPHS

  • Nam, Yun-Sun
    • Journal of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.337-344
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    • 1997
  • In this paper, we study the isomorphism problem of Cayley permutation graphs. We obtain a necessary and sufficient condition that two Cayley permutation graphs are isomrphic by a direction-preserving and color-preserving (positive/negative) natural isomorphism. The result says that if a graph G is the Cayley graph for a group $\Gamma$ then the number of direction-preserving and color-preserving positive natural isomorphism classes of Cayley permutation graphs of G is the number of double cosets of $\Gamma^\ell$ in $S_\Gamma$, where $S_\Gamma$ is the group of permutations on the elements of $\Gamma and \Gamma^\ell$ is the group of left translations by the elements of $\Gamma$. We obtain the number of the isomorphism classes by counting the double cosets.

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THE CLASSIFICATION OF SELF-DUAL CODES OF LENGTH 6 OVER ℤm FOR SMALL m

  • Park, Young Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.4
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    • pp.893-902
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    • 2010
  • In this article we study self-dual codes of length 6 over ${\mathbb{Z}}_m$. A classification of such codes for $m{\leq}24$ is given. Main tool for the classification is the new double cosets decomposition method given in the recent article of the author.