Dr. C. Mijail Borges Quintana

Departamento de Matemática
Facultad de Ciencias Naturales y Exactas
Universidad de Oriente Email: mijail@uo.edu.cu

The research group of Computer Algebra of the Universidad de Oriente, Cuba, in collaboration with the University of Valladolid, Spain, has worked for several years on the study of Gröbner representations associated with linear codes and generalizations. The ideal associated with a linear code through the equivalence relation associated with the code, offers the possibility of defining and using the Gröbner bases of these ideals for problems related to the codes, such as decoding and determining the equivalence of codes. The Gröbner bases associated with linear codes have specific properties related to the codes. This structure and generalizations such as the Gröbner representations, the set of leader codewords and the complete Gröbner bases we show that can be computed from the formalization of a specialization of a Möller-type algorithm, using the computation in the finite-dimensional quotient algebra associated with the code.

In addtion, we studied some properties and structures related to the Gröbner bases of linear codes, such as the set of leader codewords of the code, its relationship with the test sets, correctable and uncorrectable errors of general linear codes, and the construction of invariants for linear codes. We show also how some of this structures and properties can be analyzed in the more general context of group codes, like lattice codes.