Modular Forms from Codes

Maher, David P.

Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 40-58

Résumé

In this paper we construct modular forms from combinatorial designs, and codes over finite fields. We construct codes from designs, and lattices from codes. Then we use the combinatorial properties of the designs and the weight (or shape) structures of the codes to study the theta functions of the associated lattices. These theta functions are shown to be modular forms for the modular group or for various congruence subgroups. The levels of the forms we examine are determined by the dimensions of the codes and the characteristics of the fields. Using the Lee polynomial of the codes we can write the theta functions as homogeneous polynomials in modified Jacobi theta functions. By extending the underlying combinatorial structure, a modular form of half-integral weight is associated with a modular form of integral weight.

Détail

DOI : 10.4153/CJM-1980-005-6

Bibliographie

[1] 1. Assmus, E. F., Jr. and Maher, D. P., Nonexistence proofs for projective designs, to appear, American Mathematical Monthly. Google Scholar

[2] 2. Berlekamp, E. R., MacWilliams, F. J., and Sloane, N. J. A., Gleason's theorem on self-dual codes, IEEE Trans. Info. Theor. 18 (1972), 409–414. Google Scholar

[3] 3. Bourbaki, N., Groups et algebres de Lie, Ch. 4, 5, and 6 (Hermann, Paris, 1968). Google Scholar

[4] 4. Bourbaki, N., Theories spectrales (Hermann, Paris, 1967). Google Scholar

[5] 5. Broué, M., Codes correcteurs d'erreurs auto-orthogonaux sur le corps à deux elements et formes quadratiques entières définies positives à discriminant -+-1, pp. 71-108 of Comptes Rendus des Journées Mathématiques de la Société Math, de France, Univ. Sci. Tech. Languedoc, Montpellier 1974. Reprinted in Discrete Math. 17 (1977), 247–269. Google Scholar

[6] 6. Broué, M. and Enguehard, M., Polynômes des poids de certains codes et functions thêta de certains réseaux, Ann. Scient. Ec. Norm. Sup. 5 (1972), 157–181. Google Scholar

[7] 7. Cameron, P. J. and van Lint, J. H., Graph theory, coding theory, and block designs, London Math. Soc. Lecture Note Series 10 (Cambridge Univ. Press, 1975). Google Scholar

[8] 8. Conway, J. H., A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc. 1 (1969), 79–88. Google Scholar

[9] 9. Eichler, M., Introduction to the theory of algebraic numbers and functions (Academic Press, N.Y., 1966). Google Scholar

[10] 10. Gleason, A. M., Weight polynomials of self-dual codes and the MacWilliams identities in: Actes Congrès Internl. de Mathématique 3, 1970 (Gauthier-Villars, Paris, 1971), 211–215. Google Scholar

[11] 11. Leech, J.. Notes on sphere packings, Can. J. Math. 10 (1967), 251–267. Google Scholar

[12] 12. MacWilliams, F. J., Mallows, C. L., and Sloane, N. J. A., Generalizations of Gleason s theorem on weight enumerators of self-dual codes, IEEE Trans. Info. Theor. 18 (1972), 794–805. Google Scholar

[13] 13. MacWilliams, F. J., Sloane, N. J. A. and Goethals, J. M., The MacWilliams identities for nonlinear codes, Bell Syst. Tech. J. 51 (1972), 803–819. Google Scholar

[14] 14. MacWilliams, F. J., Sloane, N. J. A., and Thompson, J. G., On the existence of a projective plane of Order 10, J. Combinatorial Theory 144 (1973), 66–78. Google Scholar

[15] 15. MacWilliams, F. J. and Sloane, N. J. A., The theory of error correcting codes (North-Holland, Amsterdam, 1977). Google Scholar

[16] 16. Maher, D. P., Self-orthogonal codes and modular-forms, Ph.D. Thesis, Lehigh University, 1976. Google Scholar

[17] 17. Maher, D. P., Lee polynomials of codes and theta functions of lattices, Can. j . Math. 30 (1978), 738–747. Google Scholar

[18] 18. Niemeier, H. V., Definite Quadratische Formen der Dimension 24 und Diskcriminante 1, J. Number Theory 5 (1973), 1942–178. Google Scholar

[19] 19. Ogg, A., Modular forms and Dirichlet series (W. A. Benjamin, Inc., N. V., 1969). Google Scholar

[20] 20. Sachar, H., Error-correcting codes associated with finite projective planes, Ph.D. dissertation, Lehigh University, 1973. Google Scholar

[21] 21. Serre, J. P., Cours d'arithmétique (Presses Univ. de France, 1970). English translation published by Springer Verlag, 1973. Google Scholar

[22] 22. Shimura, G., Introduction to the arithmetic theory of automorphic functions (Princeton University Press, Princeton, N.J., 1971). Google Scholar

[23] 23. Shimura, G., On modular forms of half-integral weight, Annals of Mathematics, 2nd Ser. 97 No. 3 (1973). Google Scholar

[24] 24. Sloane, N. J. A., Codes over G F (4) and complex lattices, J. Algebr. 52 (1978), 168–181. Google Scholar

[25] 25. Tannery, J. and Molk, J., Eléments de la théorie des fonctions elliptiques, 4 vols., 2nd edition, reprinted Chelsea, N.Y. (1972). Google Scholar