On some families of invariant polynomials divisible by three and their zeta functions

TL;DR

This paper extends bounds for invariant polynomials divisible by three, introduces a new family of such polynomials unrelated to linear codes, and explores their zeta functions, advancing understanding of their algebraic and combinatorial properties.

Contribution

It establishes an analog of the Mallows-Sloane bound for Type III formal weight enumerators and identifies a new family of invariant polynomials not linked to linear codes.

Findings

Mallows-Sloane bound analog for Type III established

Existence of polynomials divisible by three, invariant under MacWilliams transform, shown

Properties of zeta functions for these polynomials discussed

Abstract

In this note, we establish an analog of the Mallows-Sloane bound for Type III formal weight enumerators. This completes the bounds for all types (Types I through IV) in synthesis of our previous results. Next we show by using the binomial moments that there exists a family of polynomials divisible by three, which are not related to linear codes but are invariant under the MacWilliams transform for the value 3/2. We also discuss some properties of the zeta functions for such polynomials.