$2$-modular lattices from ternary codes

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

2

-modular lattices from ternary codes

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

Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 73-85.

Résumé
Abstract

L’alphabet $𝐅_{3} + v 𝐅_{3}$ où $v^{2} = 1$ est vu ici comme le quotient de l’anneau des entiers du corps de nombres $𝐐 (\sqrt{- 2})$ par l’idéal (3). Les codes sur cet alphabet qui sont autoduaux pour le produit scalaire hermitien donnent des réseaux $2$ -modulaires par la construction $A_{K}$ . Il existe une application de Gray qui envoie les codes auto-duaux pour le produit scalaire euclidien sur les codes de Type III avec une involution sans points fixes dans leur groupe d’automorphismes. On démontre des théorèmes style Gleason pour les polynômes de poids symmétrisés des codes autoduaux euclidiens et pour les polynômes de poids «longueur» des codes auto-duaux hermitiens. Une application est la construction d’un réseau $2$ -modulaire optimal de dimension $18$ et de norme $3$ et de nouveaux réseaux $2$ -modulaires de norme $3$ en dimensions $16, 18, 20, 22, 24, 26, 28$ et $30$ .

The alphabet $𝐅_{3} + v 𝐅_{3}$ where $v^{2} = 1$ is viewed here as a quotient of the ring of integers of $𝐐 (\sqrt{- 2})$ 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 $30$ .

MR Zbl

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     author = {Chapman, Robin and Dougherty, Steven T. and Gaborit, Philippe and Sol\'e, Patrick},
     title = {$2$-modular lattices from ternary codes},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {73--85},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {1},
     year = {2002},
     mrnumber = {1925991},
     zbl = {1050.94018},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_1_73_0/}
}

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Chapman, Robin; Dougherty, Steven T.; Gaborit, Philippe; Solé, Patrick. $2$-modular lattices from ternary codes. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 73-85. http://www.numdam.org/item/JTNB_2002__14_1_73_0/

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