The present paper introduces a new family of linear error-correcting codes, the projective nested Cartesian codes and, using techniques of commutative algebra as Hilbert functions and Gröbner bases, studies their parameters.
Projective Cartesian codes (respectively affine Cartesian codes) of order \(d\) over a finite field \(\mathbb{F}_q\) are the image of an evaluation map of the homogeneous polynomials of degree \(d\) of \(\mathbb{F}_q[X_0,\dots,X_n]\) (respectively polynomials of \(\mathbb{F}_q[X_1,\dots,X_n]\) of degree up to \(d\)) on the points of a subset \(\chi:=A_0\times \dots \times A_n\subseteq \mathbb{P}^n(\mathbb{F}_q)\) (respectively \(\chi:=A_1\times \dots \times A_n\subseteq \mathbb{A}^n(\mathbb{F}_q)\)), \(A_i\subseteq \mathbb{F}_q\). The projective Reed-Muller codes are projective Cartesian codes with \(A_i=\mathbb{F}_q,\, \forall i\).
Section 2 gives the definition of projective nested Cartesian codes (projective Cartesian codes with additional conditions on the \(A_i\)), family including the projective Reed-Muller codes. Then formulas given the length and dimension of those codes are provided (Theorem 2.8).
Section 3 deals with the minimum distance of projective nested Cartesian codes. Lemma 3.1 gives an upper bound for that distance and Theorem 3.8 shows that the bound is the true minimum distance when the \(A_i\) are a projective nested product of fields (this is trivially the case for projective Reed-Muller codes). Finally Corollary 3.10 shows the relation between these last codes and affine Cartesian codes.