Zeta functions for formal weight enumerators and an analogue of the Mallows-Sloane bound

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

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.

$Γ$-共轭重量枚举子与不变量理论

$Γ$-conjugate weight enumerators and invariant theory

本文通过引入齐次$Γ$-共轭不变量和码的$Γ$-共轭完全重量枚举子，拓宽了码与不变量理论的联系，并证明其生成相关Clifford-Weil群的不变量环，推导出复共轭与Schur-Weyl对偶等已有结果。

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

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.

Results on zeta functions for codes

We give a new and short proof of the Mallows-Sloane upper bound for self-dual codes. We formulate a version of Greene's theorem for normalized weight enumerators. We relate normalized rank-generating polynomials to two-variable zeta functions. And we show that a self-dual code has the Clifford property, but that the same property does not hold in general for formally self-dual codes.

New asymptotic bounds for self-dual codes and lattices

We give an independent proof of the Krasikov-Litsyn bound d/n<~(1-5^{-1/4})/2 on doubly-even self-dual binary codes. The technique used (a refinement of the Mallows-Odlyzko-Sloane approach) extends easily to other families of self-dual codes, modular lattices, and quantum codes; in particular, we show that the Krasikov-Litsyn bound applies to singly-even binary codes, and obtain an analogous bound for unimodular lattices. We also show that in each case, our bound differs from the true optimum by an amount growing faster than O(n^{1/2}).