On the MacWilliams Theorem over Codes and Lattices

TL;DR

This work reframes the MacWilliams theorem in a statistical context, showing that the MacWilliams distribution over code quotients behaves like a finite Gaussian and that a smoothing parameter governs its closeness to uniform. It extends the classical identity to $m$-tuple codes via a finite Fourier-analytic approach, and demonstrates that the induced distributions on quotient spaces are statistically indistinguishable from uniform, stabilizing the finite-analytic analogy. Parallel lattice theory results are established: a Nu-function Poisson-type relation is proved for lattices arising from codes through Construction A, resolving a conjecture by Solé. Together, these results link coding theory and lattice theta structures, providing a unified probabilistic perspective with potential applications in vector quantization and cryptography. The methods fuse finite Fourier analysis on matrix rings with lattice duality to produce concrete identities tying weight enumerators, theta/nú functions, and distributional smoothing across codes and lattices.

Abstract

Analogies between codes and lattices have been extensively studied for the last decades, in this dictionary, the MacWilliams identity is the finite analog of the Jacobi-Poisson formula of the Theta function. Motivated by the random theory of lattices, the statistical significance of MacWilliams theorem is considered, indeed, MacWilliams distribution provides a finite analog of the classical Gauss distribution. In particular, the MacWilliams distribution over quotient space of a code is statistical close to the uniform distribution. In the respect of lattices, the analogy of MacWilliams identity associated with nu-function was conjectured by Sole in 1995. We give an answer to this problem in positive.