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

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 i...

Schubert varieties, linear codes and enumerative combinatorics

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have...

Enumerative combinatorics on words

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Combinatorics Enumerative Combinatorics 05

NSF Proposal: Automated Enumerative Combinatorics Automated Enumerative Combinatorics

My research students and I continued to practice a new research methodology, that can be loosely called rigorous experimental mathematics. It has something in common with both “mainstream” experimental mathematics (as preached by the Borwein brothers, David Bailey, Victor Moll, and their collaborators, see e.g. the masterpiece [BB], and the recent collection [BBCGLM]), and automated theorem pro...