Parameterized codes over some embedded sets and their applications to complete graphs (Q2869103)

Let \(X\subset \mathbb{P}^{m-1},\, \, X'\subset \mathbb{P}^{r-1},\, r<m\),\, denote two algebraic toric sets defined over the finite field \(\mathbb{F}_q\),\, with \(X'\)\, embedded in \(X\). The aim of the paper is to study the relations of the parameterized codes associated to \(X\)\, and \(X'\).NEWLINENEWLINEThe parameterized codes of order \(d\), \(\mathcal{C}_X(d)\), associated to the toric set \(X\),\, were defined by \textit{C. Rentería, A. Simis} and \textit{R. H. Villarreal} [Finite Fields Appl. 17, No. 1, 81--104 (2011; Zbl 1209.13037)]. They are the image of the evaluation map from the homogeneous polynomials of degree \(d\)\, of \(S=\mathbb{F}_q[X_1,\dots , X_m]\)\, in the points of \(X\). Section 2 relates the parameters of \(\mathcal{C}_X(d)\)\, (length, dimension and minimum distance) with the algebraic invariants of \(S/I_X,\, I_X\)\, vanishing ideal of \(X\).NEWLINENEWLINESection 3 considers the case \(X'\)\, embedded in \(X\)\, and recalls and generalizes previous results of \textit{M. Vaz Pinto} and \textit{R. H. Villarreal} [Commun. Algebra 41, No. 9, 3376--3396 (2013; Zbl 1286.13022)] relating the basic invariants of \(\mathcal{C}_X(d)\)\, and \(\mathcal{C}_X'(d)\). Section 4 provides formulas for the regularity index of \(S/I_X\)\, when \(X\)\, is the toric set parameterized by the edges of a complete graph with an even number of vertices (Corollary 5) or an odd number (Corollary 7). Finally Section 5 finds lower and upper bounds for the minimum distance of \(\mathcal{C}_X(d)\),\, \(X\)\, associated with the edges of a complete graph, both in the case of even number of vertices (Corollary 8) and odd number of vertices (Corollary 9).