On the MacWilliams Theorem Over Codes and Lattices

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