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.

Let I = I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I. ⇐) If x ∈ L 3 (C), then by Proposition 2.4, N D (x) ⊆ C \ {x}. Hence, C \ {x} is a vertex cover. A contradiction, since C is minimal. Therefore L 3 (C) = ∅.Definition 2.6. A vertex cover C of D is strong if for each x ∈ L 3 (C) there is (y, x) ∈ E(D) such that y ∈ L 2 (C) ∪ L 3 (C) with y ∈ V + (i.e. w(y) = 1).

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.

Let I be a homogeneous ideal in a polynomial ring S. In this paper, we extend the study of the asymptotic behavior of the minimum distance function δI of I and give bounds for its stabilization point, rI , when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I.

Let D be a weighted oriented graph, whose underlying graph is G, and let I(D) be its edge ideal. If G has no 3-, 5-, or 7-cycles, or G is König, we characterize when I(D) is unmixed. If G has no 3-or 5-cycles, or G is König, we characterize when I(D) is Cohen-Macaulay. We prove that I(D) is unmixed if and only if I(D) is Cohen-Macaulay when G has girth greater than 7 or G is König and has no 4-cycles.

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Let D be a weighted oriented graph, whose underlying graph is G, and let I (D) be its edge ideal. If G has no 3-, 5-, or 7-cycles, or G is Kőnig, we characterize when I (D) is unmixed. If G has no 3- or 5-cycles, or G is Kőnig, we characterize when I (D) is Cohen–Macaulay. We prove that I (D) is unmixed if and only if I (D) is Cohen–Macaulay when G has girth greater than 7 or G is Kőnig and has no 4-cycles.

Let D = (G, O, w) be a weighted oriented graph whose edge ideal is I(D). In this paper, we characterize the unmixed property of I(D) for each one of the following cases: G is an SCQ graph; G is a chordal graph; G is a simplicial graph; G is a perfect graph; G has no 4-or 5-cycles; G is a graph without 3-and 5-cycles; and girth(G) 5.