$2$-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

@article{JTNB_2002__14_1_73_0,
     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://archive.numdam.org/item/JTNB_2002__14_1_73_0/}
}

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
SP  - 73
EP  - 85
VL  - 14
IS  - 1
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_2002__14_1_73_0/
LA  - en
ID  - JTNB_2002__14_1_73_0
ER  -

%0 Journal Article
%A Chapman, Robin
%A Dougherty, Steven T.
%A Gaborit, Philippe
%A Solé, Patrick
%T $2$-modular lattices from ternary codes
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 73-85
%V 14
%N 1
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_2002__14_1_73_0/
%G en
%F JTNB_2002__14_1_73_0

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://archive.numdam.org/item/JTNB_2002__14_1_73_0/

Bibliographie
Cité par

[1] C. Bachoc, Application of coding Theory to the construction of modular lattices. J. Combin. Theory Ser. A 78 (1997),92-119. | MR | Zbl

[2] A. Bonnecaze, P. Solé, A.R. Calderbank, Quaternary quadratic residue codes and unimodular lattices. IEEE Trans. Inform. Theory 41 (1995), 366-377. | MR | Zbl

[3] A. Bonnecaze, P. Solé, C. Bachoc, B. Mourrain, Type II codes over Z4. IEEE Trans. Inform. Theory 43 (1997), 969-976. | MR | Zbl

[4] S. Buyuklieva, On the Binary Self-Dual Codes with an Automorphism of Order 2. Designs, Codes and Cryptography 12 (1) (1997), 39-48. | MR | Zbl

[5] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups. Springer, Heidelberg, 1993. | MR | Zbl

[6] J.H. Conway, N.J.A. Sloane, Self-dual codes over the integers modulo 4. J. Combin. Theory Ser. A 62 (1993), 30-45. | MR | Zbl

[7] S.T. Dougherty, Some thought about codes over groups. preprint.

[8] S.T. Dougherty, Shadow codes and weight enumerators. IEEE Trans. Inform. Theory, vol. IT-41 (1995), 762-768. | MR | Zbl

[9] S.T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, P. Solé, Self-dual Type IV codes over rings. IEEE Trans. Inform. Theory 45 (1999), 2162-2168. | MR | Zbl

[10] S.T. Dougherty, P. Gaborit, M. Harada, P. Solé, Type II codes over F2 + uF2. IEEE Trans. Inform. Theory 45 (1999), 32-45. | MR | Zbl

[11] J. Fields, P. Gaborit, J. Leon, V. Pless, All Self-Dual Z4 Codes of Length 15 or Less Are Known. IEEE Trans. Inform. Theory 44 (1998), 311-322. | MR | Zbl

[12] P. Gaborit, Mass formula for self-dual codes over Z4 and Fq + uFq rings. IEEE Trans. Inform. Theory 42 (1996), 1222-1228. | MR | Zbl

[13] M. Harada, T.A. Gulliver, H. Kaneta, Classification of extremal double circulant selfdual codes of length up to 62. Discrete Math. 188 (1998), 127-136. | MR | Zbl

[14] A.R. Hammonsjr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, P. Solé, A linear construction for certain Kerdock and Preparata codes. Bull. AMS 29 (1993), 218-222. | MR | Zbl

[15] G. Hughes, Codes and arrays from cocycles. Ph.D. thesis, Royal Melbourne Institute of Technology, 2000. | Zbl

[16] G. Hughes, Constacyclic codes, cocycles and a u+v|u-v construction. IEEE Trans. Inform. Theory 46 (2000), 674-680. | MR | Zbl

[17] F.J. Macwilliams, N.J.A. Sloane, The theory of error correcting codes. North-Holland, 1977. | Zbl

[18] J. Martinet, Les réseaux parfaits des espaces euclidiens. Masson, Paris, 1996. | MR | Zbl

[19] V. Pless, The Number of Isotropic Subspaces in a Finite Geometry. Atti. Accad. Naz. Lincei Rend. 39 (1965), 418-421. | MR | Zbl

[20] V. Pless, P. Solé, Z. Qian, Cyclic self-dual Z4-codes. Finite Fields Their Appl. 3 (1997), 48-69. | MR | Zbl

[21] H-G. Quebbemann, Modular Lattices in Euclidean Spaces. J. Number Theory 54 (1995), 190-202. | MR | Zbl

[22] E. Rains, N.J.A. Sloane, The shadow theory of modular and unimodular lattices. J. Number Theory 73 (1999), 359-389. | MR | Zbl

[23] G.C. Shephard, J.A. Todd, Finite unitary reflection groups. Can. J. Math. 6 (1954), 274-304. | MR | Zbl

[24] J. Wood, Duality for Modules over Finite Rings and Applications to Coding Theory. Amer. J. Math 121 (1999), 555-575. | MR | Zbl