Complete Gröbner basis for lattice codes

Abstract: In this work, two algorithms are developed related to lattice codes. In the first one, an extended complete Gr\"obner basis is computed for the label code of a lattice. This basis supports all term orderings associated with a total degree order offering information about de label code of the lattice. The second one is a decoding algorithm that uses an extended complete Gr\"obner basis of the label code of the lattice for monomial reduction, this provides all the lattice vectors that constitute candidates for the solution of the Close Vector Problem for a given vector.