On the MacWilliams Theorem over Codes and Lattices

TL;DR

This paper explores the analogy between codes and lattices through the MacWilliams theorem, establishing new results on the statistical properties of MacWilliams distributions and confirming a conjecture related to lattices.

Contribution

It provides a positive resolution to Sole's conjecture on the MacWilliams identity for lattices and analyzes the statistical significance of MacWilliams distributions in both contexts.

Findings

MacWilliams distribution over quotient codes is statistically close to uniform

Confirmed Sole's 1995 conjecture on MacWilliams identity for lattices

Established the finite analog of the Jacobi-Poisson formula in this setting

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.