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.

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{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 t1 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{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 “t1 2-designs” as a discrete analog of Venkov’s spherical designs of the same name.

Reçu le : 2018-12-31
Révisé le : 2019-10-14
Accepté le : 2019-10-25
Publié le : 2020-05-06
DOI : https://doi.org/10.5802/jtnb.1102
Classification : 94B05,  05B05,  11H71,  33C50
Mots clés: Type II code, extremal code, t-design, discrete harmonic polynomial
@article{JTNB_2019__31_3_679_0,
     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},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     pages = {679-688},
     doi = {10.5802/jtnb.1102},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2019__31_3_679_0/}
}
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/item/JTNB_2019__31_3_679_0/

[1] Edward F. Assmus; H. F. Mattson New 5-designs, J. Comb. Theory, Volume 6 (1969), pp. 122-151 | Article | MR 272647 | Zbl 0179.02901

[2] Christine Bachoc On harmonic weight enumerators of binary codes, Des. Codes Cryptography, Volume 18 (1999), pp. 11-28 | Article | MR 1738653 | Zbl 0961.94012

[3] John H. Conway; Vera Pless On the enumeration of self-dual codes, J. Comb. Theory, Ser. A, Volume 28 (1980), pp. 26-53 | Article | MR 558873 | Zbl 0439.94011

[4] John H. Conway; Vera Pless The binary self-dual codes of length up to 32: A revised enumeration, J. Comb. Theory, Ser. A, Volume 60 (1992), pp. 183-195 | Article | MR 1168153 | Zbl 0751.94009

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

[6] Philippe Delsarte Hahn polynomials, discrete harmonics, and t-designs, SIAM J. Appl. Math., Volume 34 (1978), pp. 157-166 | Article | MR 460158 | Zbl 0533.05009

[7] Noam D. Elkies; Scott Duke Kominers Weighted Generating Functions for Type II Lattices and Codes, Quadratic and Higher Degree Forms (Developments in Mathematics) Volume 31, Springer, 2013, pp. 63-108 | Article | MR 3156555 | Zbl 1284.94152

[8] Masaaki Harada 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 | Article | MR 2174285 | Zbl 1136.94327

[9] Masaaki Harada 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 | Article | MR 2191121 | Zbl 1142.94385

[10] Masaaki Harada 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 | Article | MR 3375567 | Zbl 1359.94700

[11] Masaaki Harada; Masaaki Kitazume; Akihiro Munemasa 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 | Article | MR 2063958 | Zbl 1048.05011

[12] Masaaki Harada; Akihiro Munemasa; Vladimir D. Tonchev A characterization of designs related to an extremal doubly-even self-dual code of length 48, Ann. Comb., Volume 9 (2005), pp. 189-198 | Article | MR 2153737 | Zbl 1076.05013

[13] Scott Duke Kominers Configurations of Extremal Even Unimodular Lattices, Int. J. Number Theory, Volume 5 (2009), pp. 457-464 | Article | MR 2529085 | Zbl 1241.11043

[14] Scott Duke Kominers Weighted Generating Functions and Configuration Results for Type II Lattices and Codes, Undergraduate Thesis, Harvard University, 2009 | Zbl 1241.11043

[15] Colin L. Mallows; Andrew M. Odlyzko; Neil J. A. Sloane Upper bounds for modular forms, lattices and codes, J. Algebra, Volume 36 (1975), pp. 68-76 | Article | MR 376536 | Zbl 0311.94002

[16] Colin L. Mallows; Neil J. A. Sloane An upper bound for self-dual codes, Inform. and Control, Volume 22 (1973), pp. 188-200 | Article | MR 414223 | Zbl 0254.94011

[17] Nathan S. Mendelsohn 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 0222.05018

[18] Michio Ozeki On even unimodular positive definite quadratic lattices of rank 32, Math. Z., Volume 191 (1986), pp. 283-291 | Article | MR 818672 | Zbl 0564.10016

[19] Michio Ozeki On the configurations of even unimodular lattices of rank 48, Arch. Math., Volume 46 (1986), pp. 54-61 | Article | MR 829816 | Zbl 0571.10020

[20] Vera Pless A classification of self-orthogonal codes over GF(2), Discrete Math., Volume 3 (1972), pp. 209-246 | Article | MR 304065 | Zbl 0256.94015

[21] Vera Pless Introduction to the Theory of Error-Correcting Codes, John Wiley & Sons, 1998 | Zbl 0928.94008

[22] Vera Pless; Neil J. A. Sloane On the Classification and Enumeration of Self-Dual Codes, J. Comb. Theory, Ser. A, Volume 18 (1975), pp. 313-335 | Article | MR 376232 | Zbl 0305.94011

[23] Jean-Pierre Serre A Course in Arithmetic, Springer, 1973 | Zbl 0256.12001

[24] Carl L. Siegel 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 252349 | Zbl 0186.08804

[25] Vladimir D. Tonchev 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 | Article | Zbl 0647.05010

[26] Boris B. Venkov Even unimodular Euclidean lattices in dimension 32, J. Math. Sci., New York, Volume 26 (1984), pp. 1860-1867 | Article | Zbl 0538.10017

[27] Boris B. Venkov 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 1878751 | Zbl 1139.11320