Complete Gr\"obner basis for lattice codes

TL;DR

This paper introduces two algorithms involving extended complete Gr"obner bases for lattice codes, enabling improved decoding and analysis of lattice structures, with applications to the Close Vector Problem.

Contribution

It presents novel algorithms for computing extended complete Gr"obner bases and applying them to lattice decoding, enhancing existing methods.

Findings

Extended complete Gr"obner basis supports all term orderings.

Decoding algorithm efficiently finds lattice vector candidates.

Algorithms improve understanding of lattice code structures.

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.