We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives an algebraic formulation of the minimum distance of a projective Reed-Muller-type code in terms of the algebraic invariants and structure of the underlying vanishing ideal. Then we give a method, based on Gröbner bases and Hilbert functions, to find lower bounds for the minimum distance of certain Reed-Mullertype codes. Finally we show explicit upper bounds for the number of zeros of polynomials in a projective nested cartesian set and give some support to a conjecture of Carvalho, Lopez-Neumann and López.We call δ I the minimum distance function of I. If I is a prime ideal, then F d = ∅ for all d ≥ 0 and δ I (d) = deg(S/I). We show that δ I generalizes the minimum distance function of projective Reed-Muller-type codes over finite fields (Theorem 4.7). This abstract algebraic formulation of the minimum distance gives a new tool to study these type of linear codes.

Abstract. Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of I i and I i stabilize for large i. We show that their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization.

In this paper we introduce the relative generalized minimum distance function (RGMDF for short) and it allows us to give an algebraic approach to the relative generalized Hamming weights of the projective Reed-Muller-type codes. Also we introduce the relative generalized footprint function and it gives a tight lower bound for the RGMDF which is much easier to compute.

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertexweighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen-Macaulay vertexweighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen-Macaulay and satisfy Alexander duality.

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Abstract. We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

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Let C be a clutter and let I be its edge ideal. We present a combinatorial description of the minimal generators of the symbolic Rees algebra Rs(I) of I. It is shown that the minimal generators of Rs(I) are in one to one correspondence with the indecomposable parallelizations of C. From our description some major results on symbolic Rees algebras of perfect graphs and clutters will follow. As a byproduct, we give a method, using Hilbert bases, to compute all indecomposable parallelizations of C and all the corresponding vertex covering numbers.