Configurations of Extremal Type II Codes via Harmonic Weight Enumerators

Elkies, Noam D.; Kominers, Scott Duke

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

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.

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.

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

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