We generalize results of Homma and Kim [J.

We study affine cartesian codes, which are a Reed-Muller type of evaluation codes, where polynomials are evaluated at the cartesian product of n subsets of a finite field F q . These codes appeared recently in a work by H. López, C. Rentería-Marquez and R. Villareal (see [11]) and, independently, in a generalized form, in a work by O. Geil and C. Thomsen (see [9]). Using a proof technique developed by O. Geil (see [8]) we determine the second Hamming weight (also called next-to-minimal weight) for particular cases of affine cartesian codes and also some higher Hamming weights of this type of code.

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

In this work we present a survey of the main results in the theory of Weierstrass semigroups at several points, with special attention to the determination of bounds for the cardinality of its set of gaps. We also review results on applications to the theory of error correcting codes. We then recall a generalization of the concept of Weierstrass semigroup, which is the Weierstrass set associated to a linear system and several points. We finish by presenting new results on this Weierstrass set, including some on the cardinality of its set of gaps.

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