On the second Hamming weight of some Reed–Muller type codes

Abstract

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.

Generalized Hamming weights of affine Cartesian codes

“…This implies that a HF J (u) ≤ a HF I (u) for all u ∈ Z. By Propositions 2.1 and 2.2, we have, Now we return to the ideal I defined in equation (1). We write I = I 1 + I 2 , where I 1 := x d1 1 , .…”

“…The codes obtained in this way are called affine cartesian codes. Affine cartesian codes were defined in [10] and further studied in, for example, [1,12,8,2]. In [10,Theorem 3.8] the authors determined the minimum distance of these codes.…”

Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes

“…Using commutative algebra tools such as regularity, degree, and Hilbert function, the authors determine the basic parameters of Cartesian codes in terms of the size of the components of the Cartesian product. In [11], the author shows some results on higher Hamming weights of Cartesian codes and gives a different proof for the minimum distance using the concepts of Gröbner basis and footprint of an ideal. In [12] the authors find several values for the second least weight of codewords, also known as the next-to-minimal Hamming weight.…”

Relative generalized Hamming weights of affine Cartesian codes

“…This generalizes the classical work [15] of Heijnen and Pelikaan towards the determination of all the generalized Hamming weights of the Reed-Muller codes. Several articles, for example [3,4], are devoted towards the determination of the next to minimal weights of affine Cartesian codes.…”