Alberto Ravagnani – University of Toronto

In coding theory, a MacWilliams identity expresses a linear relation between the weight distribution of a code and the weight distribution of the dual code. The transformation is characterized by its “Krawtchouk coefficients”.

When studying additive codes in finite abelian groups, code and dual code are subsets of different ambient spaces, and their weight enumerators refer in general to different weight functions. Invertible MacWilliams identities hold when the weights are mutually compatible. A major problem in this area is the construction of mutually compatible weights, and the computation of the associated Krawtchouk coefficients.

Using a combinatorial method, we construct a family of mutually compatible weight functions on finite abelian groups that automatically yield invertible MacWilliams identities for additive codes. The weights are obtained composing a suitable support map with the rank function of a graded (poset-) lattice having certain regularity properties. We express the corresponding Krawtchouk coefficients in terms of the combinatorial invariants of the underlying lattice, giving a closed formula for them.

The most important weight functions studied in coding theory (including the Hamming weight, the rank weight and the Lee weight) belong, up to equivalence, to the class that we introduce. This allows in particular to compute classical and new Krawtchouk coefficients employing a unified simple combinatorial method.

We conclude discussing some applications of the MacWilliams identities for the rank metric to enumerative problems of matrices.