Duality of codes supported on regular lattices, with an application to enumerative combinatorics

We construct a family of weight functions on finite abelian groups that yield invertible MacWilliams identities for additive codes. The weights are obtained composing a suitable support map with the rank function of a graded lattice that satisfies certain regularity properties. We express the Krawtchouk coefficients of the corresponding MacWilliams transformation in terms of the combinatorial invariants of the underlying lattice, and show that the most relevant weight functions studied in coding theory belong, up to equivalence, to the class that we introduce. In particular, we compute some classical Krawtchouk coefficients employing a simple combinatorial method. Our approach also allows to systematically construct weight functions that endow the underlying group with a metric space structure. We establish a Singleton-like bound for additive codes, and call optimal the codes that attain the bound. Then we prove that the dual of an optimal code is optimal, and that the weight distribution of an optimal code is completely determined by three fundamental parameters. Finally, we apply MacWilliams identities for the rank weight to enumerative combinatorics problems, computing the number of matrices of given rank over a finite field that satisfy certain linear conditions.