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

[1] Assmus, Edward F.; Mattson, H. F. New $5$-designs, J. Comb. Theory, Volume 6 (1969), pp. 122-151 | DOI | MR | Zbl

[2] Bachoc, Christine On harmonic weight enumerators of binary codes, Des. Codes Cryptography, Volume 18 (1999), pp. 11-28 | DOI | MR | Zbl

[3] Conway, John H.; Pless, Vera On the enumeration of self-dual codes, J. Comb. Theory, Ser. A, Volume 28 (1980), pp. 26-53 | DOI | MR | Zbl

[4] Conway, John H.; Pless, Vera The binary self-dual codes of length up to $32$: A revised enumeration, J. Comb. Theory, Ser. A, Volume 60 (1992), pp. 183-195 | DOI | MR | Zbl

[5] Conway, John H.; Sloane, Neil J. A. Sphere Packings, Lattices and Groups, Springer, 1999

[6] Delsarte, Philippe Hahn polynomials, discrete harmonics, and $t$-designs, SIAM J. Appl. Math., Volume 34 (1978), pp. 157-166 | DOI | MR | Zbl

[7] Elkies, Noam D.; Kominers, Scott Duke Weighted Generating Functions for Type II Lattices and Codes, Quadratic and Higher Degree Forms (Developments in Mathematics), Volume 31, Springer, 2013, pp. 63-108 | DOI | MR | Zbl

[8] Harada, Masaaki Remark on a $5$-design related to a putative extremal doubly-even self-dual $[96,48,20]$ code, Des. Codes Cryptography, Volume 37 (2005), pp. 355-358 | DOI | MR | Zbl

[9] Harada, Masaaki Self-orthogonal $3$-$(56,12,65)$ designs and extremal doubly-even self-dual codes of length $56$, Des. Codes Cryptography, Volume 38 (2006), pp. 5-16 | DOI | MR | Zbl

[10] Harada, Masaaki On a $5$-design related to a putative extremal doubly even self-dual code of length a multiple of $24$, Des. Codes Cryptography, Volume 76 (2015), pp. 373-384 | DOI | MR | Zbl

[11] Harada, Masaaki; Kitazume, Masaaki; Munemasa, Akihiro On a $5$-design related to an extremal doubly even self-dual code of length $72$, J. Comb. Theory, Ser. A, Volume 107 (2004), pp. 143-146 | DOI | MR | Zbl

[12] Harada, Masaaki; Munemasa, Akihiro; Tonchev, Vladimir D. A characterization of designs related to an extremal doubly-even self-dual code of length $48$, Ann. Comb., Volume 9 (2005), pp. 189-198 | DOI | MR | Zbl

[13] Kominers, Scott Duke Configurations of Extremal Even Unimodular Lattices, Int. J. Number Theory, Volume 5 (2009), pp. 457-464 | DOI | MR | Zbl

[14] Kominers, Scott Duke Weighted Generating Functions and Configuration Results for Type II Lattices and Codes, Undergraduate Thesis, Harvard University, 2009 | Zbl

[15] Mallows, Colin L.; Odlyzko, Andrew M.; Sloane, Neil J. A. Upper bounds for modular forms, lattices and codes, J. Algebra, Volume 36 (1975), pp. 68-76 | DOI | MR | Zbl

[16] Mallows, Colin L.; Sloane, Neil J. A. An upper bound for self-dual codes, Inform. and Control, Volume 22 (1973), pp. 188-200 | DOI | MR | Zbl

[17] Mendelsohn, Nathan S. Intersection numbers of $t$-designs, Studies in Pure Mathematics: Papers in Combinatorial Theory, Analysis, Geometry, Algebra, and the Theory of Numbers presented to Richard Rado on the Occasion of his Sixty-Fifth Birthday, Academic Press Inc., 1971, pp. 145-150 | Zbl

[18] Ozeki, Michio On even unimodular positive definite quadratic lattices of rank $32$, Math. Z., Volume 191 (1986), pp. 283-291 | DOI | MR | Zbl

[19] Ozeki, Michio On the configurations of even unimodular lattices of rank $48$, Arch. Math., Volume 46 (1986), pp. 54-61 | DOI | MR | Zbl

[20] Pless, Vera A classification of self-orthogonal codes over $\mathrm{GF}\left(2\right)$, Discrete Math., Volume 3 (1972), pp. 209-246 | DOI | MR | Zbl

[21] Pless, Vera Introduction to the Theory of Error-Correcting Codes, John Wiley & Sons, 1998 | Zbl

[22] Pless, Vera; Sloane, Neil J. A. On the Classification and Enumeration of Self-Dual Codes, J. Comb. Theory, Ser. A, Volume 18 (1975), pp. 313-335 | DOI | MR | Zbl

[23] Serre, Jean-Pierre A Course in Arithmetic, Springer, 1973 | Zbl

[24] Siegel, Carl L. Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., Volume 1969 (1969), pp. 87-102 ([pages 82–97 in Gesammelte Abhandlungen IV, Berlin: Springer 1979]) | MR | Zbl

[25] Tonchev, Vladimir D. Quasi-symmetric $2$-$(31,7,7)$ designs and a revision of Hamada’s conjecture, J. Comb. Theory, Ser. A, Volume 42 (1986), pp. 104-110 | DOI | Zbl

[26] Venkov, Boris B. Even unimodular Euclidean lattices in dimension $32$, J. Math. Sci., New York, Volume 26 (1984), pp. 1860-1867 | DOI | Zbl

[27] Venkov, Boris B. Réseaux et designs sphériques, Réseaux Euclidiens, Designs Sphériques et Formes Modulaires (Monographie de L’Enseignement Mathématique), Volume 37, L’Enseignement Mathématique, 2001, pp. 10-86 (in French) | MR | Zbl