Configurations of Extremal Type II Codes via Harmonic Weight Enumerators

Noam D. Elkies; Scott Duke Kominers

doi:10.5802/jtnb.1102

Configurations of Extremal Type II Codes via Harmonic Weight Enumerators

Noam D. Elkies ¹ ; Scott Duke Kominers ²

¹ Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138, USA
² Harvard Business School, Department of Economics, and Center of Mathematical Sciences and Applications Harvard University Rock Center for Entrepreneurship Soldiers Field, Boston, MA 02163, USA

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 679-688.

Résumé
Abstract

Nous démontrons des résultats de configuration pour les codes extrêmes de Type II analogues à ceux obtenus par Ozeki et par Kominers pour les réseaux extrêmes de Type II. Plus précisément, nous démontrons que pour

n \in {8, 24, 32, 48, 56, 72, 96}

tout code extrême de Type II et de longueur $n$ est généré par ses mots de code de poids minimal. Là où Ozeki et Kominers utilisent des harmoniques sphériques et des fonctions thêta pondérées, nous utilisons des polynômes harmoniques discrets et des énumérateurs de poids harmoniques. En cours de route, nous introduisons la notion de $t \frac{1}{2}$ -designs comme un analogue discret des dessins sphériques de Venkov portant le même nom.

We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of Kominers for extremal Type II lattices. Specifically, we show that for

n \in {8, 24, 32, 48, 56, 72, 96}

every extremal Type II code of length $n$ is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along the way we introduce “ $t \frac{1}{2}$ -designs” as a discrete analog of Venkov’s spherical designs of the same name.

Reçu le : 2018-12-30
Révisé le : 2019-10-13
Accepté le : 2019-10-24
Publié le : 2020-05-06

MR Numdam Zbl

DOI : 10.5802/jtnb.1102

Classification : 94B05, 05B05, 11H71, 33C50
Mots clés : Type II code, extremal code, $t$-design, discrete harmonic polynomial

Affiliations des auteurs :

Noam D. Elkies ¹ ; Scott Duke Kominers ²

¹ Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138, USA
² Harvard Business School, Department of Economics, and Center of Mathematical Sciences and Applications Harvard University Rock Center for Entrepreneurship Soldiers Field, Boston, MA 02163, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Noam D. Elkies and Scott Duke Kominers},
     title = {Configurations of {Extremal} {Type~II} {Codes} via {Harmonic} {Weight} {Enumerators}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {679--688},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1102},
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     mrnumber = {4102622},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1102/}
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Noam D. Elkies; Scott Duke Kominers. Configurations of Extremal Type II Codes via Harmonic Weight Enumerators. Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 679-688. doi : 10.5802/jtnb.1102. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1102/

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