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

Abstract

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The alphabet 𝐅 3 + v 𝐅 3 where v 2 = 1 is viewed here as a quotient of the ring of integers of 𝐐 ( - 2 ) by the ideal (3). Self-dual 𝐅 3 + v 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 .

How to cite

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Chapman, 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 -

References

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