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Patrick Sole - One of the best experts on this subject based on the ideXlab platform.

  • on the Algebraic Structure of quasi cyclic codes iv repeated roots
    Designs Codes and Cryptography, 2006
    Co-Authors: San Ling, Harald Niederreiter, Patrick Sole
    Abstract:

    A trace formula for quasi-cyclic codes over rings of characteristic not coprime with the co-index is derived. The main working tool is the Generalized Discrete Fourier Transform (GDFT), which in turn relies on the Hasse derivative of polynomials. A characterization of Type II self-dual quasi-cyclic codes of singly even co-index over finite fields of even characteristic follows. Implications for generator theory are shown. Explicit expressions for the combinatorial duocubic, duoquintic and duoseptic constructions in characteristic two over finite fields are given.

  • on the Algebraic Structure of quasi cyclic codes iii generator theory
    IEEE Transactions on Information Theory, 2005
    Co-Authors: San Ling, Patrick Sole
    Abstract:

    Following Parts I and II, quasi-cyclic codes of given index are studied as codes over a finite polynomial ring. These latter codes are decomposed by the Chinese Remainder Theorem (CRT), or equivalently the Mattson-Solomon transform, into products of shorter codes over larger alphabets. We characterize and enumerate self-dual one-generator quasi-cyclic codes in that context. We give an algorithm to remove some equivalent codes from that enumeration. A generalization to multigenerator codes is sketched.

  • on the Algebraic Structure of quasi cyclic codes ii chain rings
    Designs Codes and Cryptography, 2003
    Co-Authors: San Ling, Patrick Sole
    Abstract:

    The ring decomposition technique of part I is extended to the case when the factors in the direct product decomposition are no longer fields but arbitrary chain rings. This includes not only the case of quasi-cyclic codes over rings but also the case of quasi-cyclic codes over fields whose co-index is no longer prime to the characteristic of the field. A new quaternary construction of the Leech lattice is derived.

  • on the Algebraic Structure of quasi cyclic codes i finite fields
    IEEE Transactions on Information Theory, 2001
    Co-Authors: San Ling, Patrick Sole
    Abstract:

    A new Algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese remainder theorem (CRT), or of the discrete Fourier transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the (u+/spl upsi/|u-/spl upsi/) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes.

Diganta Saha - One of the best experts on this subject based on the ideXlab platform.

Murad Alim - One of the best experts on this subject based on the ideXlab platform.

  • Algebraic Structure of tt equations for calabi yau sigma models
    Communications in Mathematical Physics, 2017
    Co-Authors: Murad Alim
    Abstract:

    The tt * equations define a flat connection on the moduli spaces of $${2d, \mathcal{N}=2}$$ quantum field theories. For conformal theories with c = 3d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the tt * equations, restricted to the conformal directions, in the cases d = 1, 2, 3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with the generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of tt * and provides these with a finer and Algebraic meaning. For sigma models into lattice polarized K3 manifolds, the differential ring of special functions on the moduli space is constructed, extending known Structures for d = 1 and 3. The generators of the differential rings of special functions are given by quasi-modular forms for d = 1 and their generalizations in d = 2, 3. Some explicit examples are worked out including the case of the mirror of the quartic in $${\mathbbm{P}^3}$$ , where due to further Algebraic constraints, the differential ring coincides with quasi modular forms.

  • Algebraic Structure of tt equations for calabi yau sigma models
    arXiv: High Energy Physics - Theory, 2014
    Co-Authors: Murad Alim
    Abstract:

    The $tt^*$ equations define a flat connection on the moduli spaces of $2d, \mathcal{N}=2$ quantum field theories. For conformal theories with $c=3d$, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the $tt^*$ equations in the cases $d=1,2,3$ is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of $tt^*$ and provides these with a finer and Algebraic meaning. For sigma models into lattice polarized $K3$ manifolds, the differential ring of special functions on the moduli space is constructed, extending known Structures for $d=1$ and 3. The generators of the differential rings of special functions are given by quasi-modular forms for $d=1$ and their generalizations in $d=2,3$. Some explicit examples are worked out including the case of the mirror of the quartic in $CP^3$, where due to further Algebraic constraints, the differential ring coincides with quasi modular forms.

San Ling - One of the best experts on this subject based on the ideXlab platform.

  • on the Algebraic Structure of quasi cyclic codes iv repeated roots
    Designs Codes and Cryptography, 2006
    Co-Authors: San Ling, Harald Niederreiter, Patrick Sole
    Abstract:

    A trace formula for quasi-cyclic codes over rings of characteristic not coprime with the co-index is derived. The main working tool is the Generalized Discrete Fourier Transform (GDFT), which in turn relies on the Hasse derivative of polynomials. A characterization of Type II self-dual quasi-cyclic codes of singly even co-index over finite fields of even characteristic follows. Implications for generator theory are shown. Explicit expressions for the combinatorial duocubic, duoquintic and duoseptic constructions in characteristic two over finite fields are given.

  • on the Algebraic Structure of quasi cyclic codes iii generator theory
    IEEE Transactions on Information Theory, 2005
    Co-Authors: San Ling, Patrick Sole
    Abstract:

    Following Parts I and II, quasi-cyclic codes of given index are studied as codes over a finite polynomial ring. These latter codes are decomposed by the Chinese Remainder Theorem (CRT), or equivalently the Mattson-Solomon transform, into products of shorter codes over larger alphabets. We characterize and enumerate self-dual one-generator quasi-cyclic codes in that context. We give an algorithm to remove some equivalent codes from that enumeration. A generalization to multigenerator codes is sketched.

  • on the Algebraic Structure of quasi cyclic codes ii chain rings
    Designs Codes and Cryptography, 2003
    Co-Authors: San Ling, Patrick Sole
    Abstract:

    The ring decomposition technique of part I is extended to the case when the factors in the direct product decomposition are no longer fields but arbitrary chain rings. This includes not only the case of quasi-cyclic codes over rings but also the case of quasi-cyclic codes over fields whose co-index is no longer prime to the characteristic of the field. A new quaternary construction of the Leech lattice is derived.

  • on the Algebraic Structure of quasi cyclic codes i finite fields
    IEEE Transactions on Information Theory, 2001
    Co-Authors: San Ling, Patrick Sole
    Abstract:

    A new Algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese remainder theorem (CRT), or of the discrete Fourier transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the (u+/spl upsi/|u-/spl upsi/) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes.

Sandip Paul - One of the best experts on this subject based on the ideXlab platform.