Abstract :
The old problem of counting lattice points in euclidean spheres leads to use Jacobi theta functions and its relatives as generating functions. Important lattices (root systems, the Leech lattice) can be constructed from algebraic codes and analogies between codes and lattices have been extensively studied by coding theorists and number theorists alike. In this dictionary, the MacWilliams formula is the finite analog of the Poisson formula.
The new problem of counting lattice points in spheres for the L1 distance leads to hyperbolic trigonometric functions. The same analogy exists but the L1 counterpart of the Poisson formula is missing. The MacWilliams formula leads to such a duality formula for those lattices which are constructed from codes via Construction A. A connection with Ehrart enumerative theory of polytopes is pointed out. Both problems have important applications in multidimensional vector quantization.
