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Modular Forms from Codes

Published online by Cambridge University Press:  20 November 2018

David P. Maher
David P. Maher*
Affiliation:
Worcester Polytechnic Institute, Worcester, Massachusetts
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 40 - 58
Copyright
Copyright © Canadian Mathematical Society 1980

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