MacWilliams Theory over Zk and nu-functions over Lattices

TL;DR

The paper generalizes MacWilliams duality to $m$-tuple weight enumerators over $\mathbb{Z}_k$ and to complete weight enumerators over $\mathbb{Z}_k[xi]$, establishing the corresponding duality identities $W_{C^{perp}}^{(m)}(z) = (1/|C|) (1+(k^m-1) z)^n W_{C}^{(m)}( (1-z)/(1+(k^m-1) z) )$ for codes and its multilinear variant. It then extends finite Fourier transform techniques to the matrix ring $\mathbb{Z}_k^{m\times n}$ and proves the generalized Poisson summation leading to the $m$-tuple MacWilliams identity for code families. The study of the nu-function on lattices yields a new ternary-code lattice formula (Theorem 4) and demonstrates, via counterexamples, that Solé's conjecture linking dual nu-functions fails in general, though it holds for lattices associated with binary codes. Collectively, the results deepen the code–lattice duality theory, clarify the limits of conjectured continuous dualities in higher algebraic structures, and have implications for multidimensional vector quantization and lattice-based coding schemes.

Abstract

Continuing previous works on MacWilliams theory over codes and lattices, a generalization of the MacWilliams theory over $\mathbb{Z}_k$ for $m$ codes is established, and the complete weight enumerator MacWilliams identity also holds for codes over the finitely generated rings $\mathbb{Z}_k[ξ]$. In the context of lattices, the analogy of the MacWilliams identity associated with nu-function was conjectured by Solé in 1995, and we present a new formula for nu-function over the lattices associated with a ternary code, which is rather different from the original conjecture. Furthermore, we provide many counterexamples to show that the Solé conjecture never holds in the general case, except for the lattices associated with a binary code.