Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields (Q620935)

Let \(\mathbb{K}\) be a field and let \(\mathbb{K}^*\) denote its multiplicative group. Let \(\scriptstyle D=\left(d_{ij}\right)_{0\leq i\leq m-1}^{0\leq j\leq n-1}\in\mathbb{Z}^{m\times n}\) be an \(m\times n\) matrix with integer entries. For a collection of \(n\) variables \(\scriptstyle{\mathbf X}=\{X_0,\ldots,X_{n-1}\}\), let \(M_D = \left\{{\mathbf X}^{{\mathbf d}_i} = \prod_{j=0}^{n-1}X^{d_{ij}}\right\}_{i=0}^{m-1}\) be the collection of Laurent monomials determined by \(D\). The corresponding algebraic toric set is \(P_D = \left\{ \left({\mathbf x}^{{\mathbf d}_0},\ldots,{\mathbf x}^{{\mathbf d}_{m-1}}\right)\in\mathbb{K}^m|\;{\mathbf x}\in(\mathbb{K}^*)^n\right\}\) which is indeed a subset of the projective space \(\mathbb{P}^{m-1}\). The vanishing ideal \(I(P_D)\) is the ideal generated by the homogeneous polynomials in \(\mathbb{K}[{\mathbf X}]\) that vanish on \(P_D\). The authors analyze the ideal \(I(P_D)\) and they show that it is a radical Cohen-Macaulay lattice ideal of dimension 1, they provide a list of generators in a ring extension and they develop some methods to compute the Hilbert function of the quotient \(\mathbb{K}[{\mathbf X}]/I(P_D)\) with respect to the ring gradation given by the polynomial total degree. For a fixed degree \(d\in\mathbb{Z}^+\), let \(\mathbb{K}[{\mathbf X}]_d\) denote the subset consisting of degree \(d\) polynomials. In the Reed-Muller style, let \(\scriptstyle\text{ev}_d:\mathbb{K}[{\mathbf X}]_d\to\mathbb{K}^{n_D}\), \(P({\mathbf X})\mapsto\text{ev}_d(P({\mathbf X})) = \left(\frac{1}{x_{0\nu}^d}P({\mathbf x}_{\nu})\right)_{\nu=0}^{n_D-1}\) be an evaluation map, where \(n_D=\text{card}(P_D)\). The image of this map is a linear code \(C_D(d)\), of length \(n_D\), dimension \(k_D(d)\) and minimum distance \(\delta_D(d)\). The authors present several methods, using the Hilbert function, to estimate the code parameters \(n_D\), \(k_D(d)\) and \(\delta_D(d)\), including the cases in which the built code \(C_D(d)\) arises from connected non-bipartite graphs and the cases in which the Singleton upper bound for the minimum distance is attained. The paper is a rather complete and detailed exposition of the methods and it points out several partial results at which the use of tools for symbolic computation were fruitfully exploited.