Theta Functions and Self-Dual Codes

Abstract

There is a connection between binary linear codes and lattices in n-space. To every lattice is associated its theta series. The Poisson summation formula gives a relation between the theta series of a lattice and that of its dual. For even unimodular lattices this tells us that the theta series is a modular form. Classical relations among modular forms then yield results about lattices and the associated linear codes. In this chapter we shall give an introduction to this fascinating relationship between seemingly disparate topics. In particular, we will give another proof for the MacWilliams identity. Our exposition in this chapter is based on [7]. A wealth of information and inspiration can be found in [4].