Linear label code of a root lattice using Gröbner bases

The label code of a lattice plays a key role in the characterization of the lattice. Every lattice \(\Lambda\) can be described in terms of a label code L and an orthogonal sublattice \(\Lambda '\) such that \(\Lambda /\Lambda '\cong L\). We identify the binomial ideal associated to an integer lattice and then establish a relation between the ideal quotient of the lattice and its label code. Furthermore, we present the Gröbner basis of the well-known root lattice \(D_n\). As an application of the relation \(I_{\Lambda }=I_{\Lambda '}+I_{L}\), where \(I_{\Lambda },I_{\Lambda '}\) and \(I_L\) denote binomial ideals associated to \(\Lambda ,~\Lambda '\) and L, respectively, a linear label code of \(D_n\) is obtained using its Gröbner basis.