  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.
