This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce q-polymatroids, the q-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.We start by establishing the notation and the definitions used throughout the paper.Notation 1.1. In the sequel, we fix integers n, m ≥ 2 and a prime power q. For an integer t, we let [t] := {1, ..., t}. We denote by F q the finite field with q elements. The space of n × m matrices with entries in F q is denoted by Mat. Up to transposition, we assume without loss of generality that n ≤ m. We let Mat(J, c) = {M ∈ Mat | colsp(M ) ⊆ J} and Mat(J, r) = {M ∈ Mat | rowsp(M ) ⊆ J}.Throughout the paper, we only consider linear codes. All dimensions are computed over F q , unless otherwise stated.

We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with prescribed parameters of a certain type. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.

In this paper we introduce a new family of codes, called projective nested cartesian codes. They are obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of P n (F q ), and they may be seen as a generalization of the so-called projective Reed-Muller codes. We calculate the length and the dimension of such codes, an upper bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed-Muller codes). At the end we show some relations between the parameters of these codes and those of the affine cartesian codes.1 2 CÍCERO CARVALHO, V. G. LOPEZ NEUMANN, AND HIRAM H. LÓPEZ defines a linear map of K-vector spaces. The image of ϕ d , denoted by C X (d), defines a linear code (as usual by a linear code we mean a linear subspace of K |X | ). We call C X (d) a projective cartesian code of order d defined over A 0 , . . . , A n . Thus the projective cartesian codes are part of the family of evaluation codes defined on a subset of a projective space, see [4], [5], [7] and [10] for other examples. An important special case of the projective cartesian codes, which served as motivation for our work, is the one where A i = K for all i = 0, . . . , n. Then we have X = P n and C X (d) is the so-called projective Reed-Muller code (of order d), as defined and studied in [11] or [14].The dimension and the length of C X (d) are given by dim K C X (d) (dimension as Kvector space) and |X | respectively. The minimum distance of C X (d) is given bywhere |ϕ d (f )| is the number of non-zero entries of ϕ d (f ). These are the main parameters of the code C X (d) and they are presented in the main results of this paper, although we find the minimum distance only when the A ′ i s satisfy certain conditions (Definition 2.1). In the next section we compute the length and the dimension of C X (d), and to do this we use some concepts of commutative algebra which we now recall. The vanishing ideal of X ⊂ P n , denoted by I(X ), is the ideal of S generated by the homogeneous polynomials that vanish on all points of X . We are interested in the algebraic invariants (degree, Hilbert function) of I(X ), because the kernel of the evaluation map, ϕ d , is precisely I(X ) d , where I(X ) d := S d ∩ I(X ). In general, for any subset (ideal or not) F of S we define F d := F ∩ S d . The Hilbert function of S/I(X ) is given byso H X (d) is precisely the dimension of C X (d).We will also need tools from Gröbner bases theory, which we recall briefly.Let ≺ be a monomial order defined on the set M of monomials of the polynomial ring S, i.e. ≺ is a total order on M, we have 1 ≺ M for any monomial M, and if M 1 ≺ M 2 then MM 1 ≺ MM 2 for all M, M 1

Let K be a finite field and let X * be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Gröbner bases, to compute the length and the dimension of CX * (d), the parameterized affine code of degree d on the set X * . If Y is the projective closure of X * , it is shown that CX * (d) has the same basic parameters that CY (d), the parameterized projective code on the set Y . If X * is an affine torus, we compute the basic parameters of CX * (d). We show how to compute the vanishing ideals of X

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