# $2$-modular lattices from ternary codes

Robin Chapman; Steven T. Dougherty; Philippe Gaborit; Patrick Solé

Journal de théorie des nombres de Bordeaux (2002)

- Volume: 14, Issue: 1, page 73-85
- ISSN: 1246-7405

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topChapman, Robin, et al. "$2$-modular lattices from ternary codes." Journal de théorie des nombres de Bordeaux 14.1 (2002): 73-85. <http://eudml.org/doc/248902>.

@article{Chapman2002,

abstract = {The alphabet $\mathbf \{F\}_3 + v\mathbf \{F\}_3$ where $v^2 = 1$ is viewed here as a quotient of the ring of integers of $\mathbf \{Q\}(\sqrt\{-2\})$ by the ideal (3). Self-dual $\mathbf \{F\}_3 + v\mathbf \{F\}_3$ codes for the hermitian scalar product give $2$-modular lattices by construction $A_K$. There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian self-dual codes are derived. As an application we construct an optimal $2$-modular lattice of dimension $18$ and minimum norm $3$ and new odd $2$-modular lattices of norm $3$ for dimensions $16,18,20,22,24,26,28$ and $30$.},

author = {Chapman, Robin, Dougherty, Steven T., Gaborit, Philippe, Solé, Patrick},

journal = {Journal de théorie des nombres de Bordeaux},

language = {eng},

number = {1},

pages = {73-85},

publisher = {Université Bordeaux I},

title = {$2$-modular lattices from ternary codes},

url = {http://eudml.org/doc/248902},

volume = {14},

year = {2002},

}

TY - JOUR

AU - Chapman, Robin

AU - Dougherty, Steven T.

AU - Gaborit, Philippe

AU - Solé, Patrick

TI - $2$-modular lattices from ternary codes

JO - Journal de théorie des nombres de Bordeaux

PY - 2002

PB - Université Bordeaux I

VL - 14

IS - 1

SP - 73

EP - 85

AB - The alphabet $\mathbf {F}_3 + v\mathbf {F}_3$ where $v^2 = 1$ is viewed here as a quotient of the ring of integers of $\mathbf {Q}(\sqrt{-2})$ by the ideal (3). Self-dual $\mathbf {F}_3 + v\mathbf {F}_3$ codes for the hermitian scalar product give $2$-modular lattices by construction $A_K$. There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian self-dual codes are derived. As an application we construct an optimal $2$-modular lattice of dimension $18$ and minimum norm $3$ and new odd $2$-modular lattices of norm $3$ for dimensions $16,18,20,22,24,26,28$ and $30$.

LA - eng

UR - http://eudml.org/doc/248902

ER -

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