On some families of divisible formal weight enumerators and their zeta functions

TL;DR

This paper investigates three families of divisible formal weight enumerators, exploring their properties such as bounds, extremal characteristics, and Riemann hypothesis analogs, while generalizing invariant differential operator theory.

Contribution

It introduces new families of divisible formal weight enumerators and extends the theory of invariant differential operators related to their properties.

Findings

Analogues of Mallows-Sloane bounds established

Extremal properties of the families analyzed

Connections to Riemann hypothesis-like conditions

Abstract

The formal weight enumerators were first introduced by M. Ozeki, and it was shown in the author's previous paper that there are various families of divisible formal weight enumerators. Among them, three families are dealt with in this paper and their properties are investigated: they are analogs of the Mallows-Sloane bound, the extremal property, the Riemann hypothesis, etc. In the course of the investigation, some generalizations of the theory of invariant differential operators developed by I. Duursma and T. Okuda are deduced.