Weighted Generating Functions and Configuration Results for Type II Lattices and Codes

We present an exposition of weighted theta functions, which are weighted generating functions for the norms and distribution of lattice vectors. We derive a decomposition theorem for the space of degree-d homogeneous polynomials in terms of spaces of harmonic polynomials and then prove that the weighted theta functions of Type II lattices are examples of modular forms. Our development of these results is structural, related to the infinite-dimensional representation theory of the Lie algebra sl2. We give several applications of weighted theta functions: a condition on the root systems of Type II lattices of rank 24; a proof that extremal Type II lattices yield spherical t-designs; and configuration results for extremal Type II lattices of ranks 8, 24, 32, 40, 48, 56, 72, 80, 96, and 120, one of which has not appeared previously. Then, we give a new structural development of harmonic weight enumerators—the codingtheoretic analogs of weighted theta functions—in analogy with our approach to weighted theta functions. We use the finite-dimensional representation theory of sl2 to derive a decomposition theorem for the space of degree-d discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials and then prove a generalized MacWilliams identity for harmonic weight enumerators. Next, we present several applications of harmonic weight enumerators analogous to those given for weighted theta functions: an equivalent characterization of t-designs and the extremal Type II code case of the Assmus–Mattson Theorem; a condition on the tetrad systems of Type II codes of length 24; and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96. Nearly all of these applications are original to this thesis, and many explicitly use components of our development of the harmonic weight enumerator theory.