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

Let K be a finite field, let X ⊂ ℙ^m-1 and X′ ⊂ ℙ^r-1, with r < m, be two algebraic toric sets parameterized by some monomials in such a way that X′ is embedded in X. We describe the relations among the main parameters of the corresponding parameter-ized linear codes of order d associated to X and X′ by using some tools from commutative algebra and algebraic geometry. We also find the regularity index in the case of toric sets parameterized by the edges of a complete graph. Finally, we give some bounds for the minimum distance of the linear codes associated to complete graphs.