Configurations of Extremal Type II Codes

TL;DR

This paper establishes that extremal Type II codes of specific lengths are generated by their minimal weight codewords, using discrete harmonic polynomials and introducing new design concepts.

Contribution

It proves generation results for extremal Type II codes at certain lengths using discrete harmonic analysis, extending previous lattice-based results.

Findings

Extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, 96 are generated by minimal weight codewords.

Introduces '$t\frac12$-designs' as a discrete analog of spherical designs.

Uses discrete harmonic polynomials and harmonic weight enumerators for analysis.

Abstract

We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for $n \in \{8, 24, 32, 48, 56, 72, 96\}$ every extremal Type II code of length $n$ is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along we way we introduce "$t\frac12$-designs" as a discrete analog of Venkov's spherical designs of the same name.