Weighted Generating Functions for Type II Lattices and Codes

TL;DR

This paper develops a new theoretical framework for harmonic polynomials and weight enumerators in binary codes using representation theory, leading to new identities and applications in design theory and extremal codes.

Contribution

It introduces a novel decomposition of harmonic polynomials in coding theory using sl_2 representation theory and generalizes MacWilliams identities for harmonic weight enumerators.

Findings

Derived a decomposition theorem for discrete harmonic polynomials.

Proved a generalized MacWilliams identity for harmonic weight enumerators.

Applied results to characterize t-designs and analyze extremal Type II codes.

Abstract

We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean lattices. Namely, we use the finite-dimensional representation theory of sl_2 to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weight enumerators, corresponding to some uses of weighted theta functions: an equivalent characterization of t-designs, the Assmus-Mattson Theorem in the case of extremal Type II codes, and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96.