Projective Nested Cartesian Codes

Abstract

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

Minimum distance functions of graded ideals and Reed–Muller-type codes

“…As an application, Corollary 4.9 will be used to study the minimum distance of projective nested cartesian codes [3] over a set X . In this case t i is a zero-divisor of S/I(X ) for i = 1, .…”

“…Projective nested cartesian codes were introduced and studied in [3] (see Definition 6.1). This type of evaluation codes generalize the classical projective Reed-Muller codes [21].…”

“…al. found a Gröbner basis for I(X ) whose initial ideal is L, and obtained formulas for the regularity and the degree of the coordinate ring S/I(X ) [3] (Proposition 6.3).…”

“…They showed the conjecture when the A i 's are subfields of F q , and essentially showed that their conjecture can be reduced to: Conjecture 6.4. (Carvalho, Lopez-Neumann, and López [3]…”

Generalized minimum distance functions

“…The Reed-Muller-type codes and their parameters have been studied extensively. If X is a subset of a projective space P s´1 over a finite field K " F q , and C X pdq is the corresponding Reed-Muller-type code (Definition 2.5), several cases have been described [1], [3], [7], [8], [9], [10], [11], [12], [14], [15], [19], [21], [25], [26], [27], [28], [29], [30], [31]:…”