Semidefinite programming, harmonic analysis and coding theory (2009)
These lecture notes where presented as a course of the CIMPA summer school in Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics.
Semidefinite programming, harmonic analysis and coding theory (2009)
These lecture notes where presented as a course of the CIMPA summer school in Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics.
Semidefinite programming, harmonic analysis and coding theory (2009)
These lecture notes where presented as a course of the CIMPA summer school in Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics.
On Principal Units of Qp(ζp (2008)
Miriam Abdón, Christine Bachoc
Arithmétique, géométrie et théorie des codes Arithmetic, geometry and coding theory
On Principal Units of Qp(ζp (2008)
Miriam Abdón, Christine Bachoc
Arithmétique, géométrie et théorie des codes Arithmetic, geometry and coding theory
Abstract — Upper bounds are derived for codes in the Grassmann manifold with given minimum chordal distance. They stem from upper bounds for codes in the product of unit spheres and projective...
Bounds for Codes in Products of Spaces, Grassmann and Stiefel Manifolds (2008)
Christine Bachoc, Yael Ben-haim, Simon Litsyn, Senior Member
Upper bounds are derived for codes in Stiefel and Grassmann manifolds with given minimum chordal distance. They stem from upper bounds for codes in the product of unit spheres and projective spaces....
OPTIMALITY AND UNIQUENESS OF THE (4,10,1/6) SPHERICAL CODE (2008)
Christine Bachoc, Frank Vallentin
ABSTRACT. Traditionally, optimality and uniqueness of an (n, N, t) spherical code is proved using linear programming bounds. However, this approach does not apply to the parameter (4, 10, 1/6). We...
Lower bounds for measurable chromatic numbers (2008)
Bachoc, Christine, Nebe, Gabriele, Vallentin, Frank
The Lovasz theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower...
Designs, groups and lattices (2007)
We study the Grassmannian 4-designs contained in lattices, in connection with the local property of the Rankin constant. We prove that the sequence of Barnes-Wall lattices contain Grassmannian...
Christine Bachoc, Renaud Coulangeon, Gabriele Nebe
Abstract. We introduce the notion of a t-design on the Grassmann manifold Gm;n of the m-subspaces of the Euclidean space R
Zonal functions for the unitary groups and applications to hermitian lattices (2007)
Christine Bachoc, Gabriele Nebe
) under the actions of the complex and quaternionic unitary groups. We give an explicit basis for the space of zonal functions, which in the second case takes account of the action of the group of...
Christine Bachoc, T. Aaron Gulliver, Masaaki Harada
A code is called isodual if it is equivalent to its dual code, and a lattice is called isodual if it is isometric to its dual lattice. In this note, we investigate isodual codes over Z 2k. These...
On extremal additive GF (4)-codes of lengths 10 to 18 (2007)
Abstract. In this paper we classify the 19 even extremal self-dual additive codes of length 10 and we classify under a restrictive hypothesis 490 even extremal self-dual codes of length 14. Under the...
Odd unimodular lattices of minimum 4 (2007)
Christine Bachoc, Gabriele Nebe, Boris Venkov
We prove the non existence of unimodular lattices of minimum 4 and dimension 34 and 35. 1
Macwilliams Identities, Christine Bachoc
Abstract. We dene harmonic weight enumerators asociated to codes dened over a group alphabet F of size q. They generalize the classical Hamming weight enumerator and are associated to the...
Etude Algorithmique de R eseaux Construits avec la Forme Trace (2007)
Christine Bachoc, Christian Batut, Table Des Matieres
2. R eseaux unimodulaires en dimension 24, 32 et 48 3. Certains r eseaux li es au r eseau de Leech 4. Les r eseaux de Craig
Zonal Functions for the Unitary Groups and Applications to Hermitian Lattices (2007)
Christine Bachoc, Gabriele Nebe
We study the decomposition of the space L ) under the actions of the complex and quaternionic unitary groups. We give an explicit basis for the space of zonal functions, which in the second case...
Optimality and uniqueness of the (4,10,1/6) spherical code (2007)
Bachoc, Christine, Vallentin, Frank
Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use...
Forme Trace et Ramification Sauvage (2006)
Let A(K/Q) denote the fractional ideal of a cyclic p-extension K/Q whose square is the inverse different of the extension. Equipped with the trace form, A(K/Q) becomes a Z Gal (K/Q)-hermitian module...
Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps (2006)
Bachoc, Christine, Vallentin, Frank
We apply the semidefinite programming approach developed in arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps. We compute new upper bounds for the one-sided kissing number...
Linear programming bounds for codes in Grassmannian spaces (2006)
We introduce a linear programming method to obtain bounds on the cardinality of codes in Grassmannian spaces for the chordal distance. We obtain explicit bounds, and an asymptotic bound that improves...
Bounds for codes in products of spaces, Grassmann and Stiefel manifolds (2006)
Bachoc, Christine, Ben-Haim, Yael, Litsyn, Simon
Upper bounds are derived for codes in Stiefel and Grassmann manifolds with given minimal chordal distance. They stem from upper bounds for codes in products of unit spheres and projective spaces. The...
New upper bounds for kissing numbers from semidefinite programming (2006)
Bachoc, Christine, Vallentin, Frank
Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for...
On theta series vanishing at infinity and related lattices (2006)
Bachoc, Christine, Salvati Manni, Riccardo
We consider theta series with the highest possible order of vanishing at infinity when the level is a power of 2 or 3 and the lattices associated to these theta series. We prove that these lattices...
On theta series vanishing at infinity and related lattices (2006)
Bachoc, Christine, Salvati Manni, Riccardo
We consider theta series with the highest possible order of vanishing at infinity when the level is a power of 2 or 3 and the lattices associated to these theta series. We prove that these lattices...
Linear programming bounds for codes in Grassmannian spaces (2006)
ABSTRACT. We develop the linear programming method to obtain bounds for the cardinality of Grassmannian codes endowed with the chordal distance. We obtain a bound and its asymptotic version that...
New upper bounds for kissing numbers from semidefinite programming (2006)
Christine Bachoc, Frank Vallentin
In geometry, the kissing number problem asks for the maximum number τn of unit spheres that can simultaneously touch the unit sphere in n-dimensional Euclidean space without pairwise overlapping....
Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps (2006)
Christine Bachoc, Frank Vallentin
ABSTRACT. In this paper we apply the semidefinite programming approach developed in [2] to obtain new upper bounds for codes in spherical caps. We compute new upper bounds for the one-sided kissing...
New upper bounds for kissing numbers from semidefinite programming (2006)
Christine Bachoc, Frank Vallentin
ABSTRACT. Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper...
Designs, groups and lattices (2005)
We study the Grassmannian 4-designs contained in lattices, in connection with the local property of the Rankin constant. We prove that the sequence of Barnes-Wall lattices contain Grassmannian...
Designs, groups and lattices (2005)
We study the Grassmannian 4-designs contained in lattices, in connection with the local property of the Rankin constant. We prove that the sequence of Barnes-Wall lattices contain Grassmannian...
In Low Dimensions, Frank Vallentin, Wien Österreich, Prof Dr, Christine Bachoc, ...
zur Erlangung des akademischen Grades eines genehmigten Dissertation. Doktors der Naturwissenschaften Vorsitzender: Univ.-Prof. Dr. PETER GRITZMANN
Designs And Self-Dual Codes With Long Shadows (2002)
Christine Bachoc, Philippe Gaborit
In this paper we introduce the notion of s-extremal codes for self-dual binary codes and we relate this notion to the existence of 1-designs or 2-designs in these codes. We extend the classification...
Designs in Grassmannian Spaces and Lattices (2002)
Christine Bachoc, Renaud Coulangeon, Gabriele Nebe
We introduce the notion of a t-design on the Grassmann manifold Gm;n of the m-subspaces of the Euclidean space R . It generalizes the notion of antipodal spherical design which was introduced by P....
Siegel Modular Forms, Grassmannian Designs, (2002)
And Unimodular Lattices, Christine Bachoc, Gabriele Nebe
Siegel theta series with harmonic coecients are vectorvalued Siegel modular forms. We use them to show that certain sections of lattices form designs in Grassmannian space.
Christine Bachoc, Boris Venkov
With the help of modular forms, we compute some Jacobi forms associated to n-dimensional extremal lattices of prime level l and determinant l n=2. We show that the layers of these lattices support...
Odd unimodular lattices of minimum 4 (2000)
Christine Bachoc, Gabriele Nebe, Boris Venkov
Abstract We prove the non existence of unimodular lattices of minimum 4 and dimension 34 and 35. 1 Introduction Unimodular lattices have focus interest for a long time. One of the most fascinating...
On Harmonic weight enumerators of binary codes (1999)
We dene some new polynomials associated to a linear binary code and a harmonic function of degree k. The case k = 0 is the usual weight enumerator of the code. When divided by (xy) k, they satisfy a...
Extremal lattices of minimum 8 related to the Mathieu group M 22 (1998)
Christine Bachoc, Gabriele Nebe
In this paper, we construct three new extremal lattices of minimum 8; one is 3-modular and of dimension 40, the two others are unimodular of dimension 80. They are strongly connected to the...
Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order (1997)
Bachoc, Christine, Nebe, Gabriele
A generalization of Kneser's neighboring method allows us to classify two interesting genera at the same time. The new method is used to determine the genus of Hermitian unimodular lattices of rank 8...
Applications of coding theory to the construction of modular lattices (1997)
Abstract. We study self-dual codes over certain finite rings which are quotients of quadratic imaginary fields or of totally definite quaternion fields over Q. A natural weight taking two different...
Alexis Bonnecaze, Patrick Solé, Christine Bachoc, Bernard Mourrain
Type II Z 4 -codes are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with euclidean weights multiple of 8: Their weight enumerators are characterized by...
Étude algorithmique de réseaux construits avec la forme trace (1992)
Bachoc, Christine, Batut, Christian
We study the numerical properties of three types of lattices constructed by means of the trace form in cyclotomic number fields. We calculate their minimum and minimal vectors, and determine whether...
Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order.
Christine Bachoc, Gabriele Nebe
this paper we generalize this method, replacing the maximal sublattices by sublattices of larger index in a more interesting genus. The resulting graph in each genus, which factors over a bipartite...