Generalized Hamming weights of affine Cartesian codes

Abstract

In this article, we give the answer to the following question: Given a field F, finite subsets A 1 , . . . , Am of F, and r linearly independent polynomials f 1 , . . . , fr ∈ F[x 1 , . . . , xm] of total degree at most d. What is the maximal number of common zeros f 1 , . . . , fr can have in A 1 ×· · ·×Am? For F = Fq, the finite field with q elements, answering this question is equivalent to determining the generalized Hamming weights of the so-called affine Cartesian codes. Seen in this light, our work is a generalization of the work of Heijnen-Pellikaan for Reed-Muller codes to the significantly larger class of affine Cartesian codes.

Relative generalized Hamming weights of affine Cartesian codes

“…We are now ready to state and prove the main theorem of this section. This is a generalization of [1,Theorem 3.8]. Further special cases, when d 1 = · · · = d m = q, appear as [21,Lemma 6], [15,Theorem 5.7] and [11,Lemma 4.6].…”

“…(2) in general, without any constraint on u 2 , the question corresponds to the determination of the RGHWs of the Reed-Muller codes, and as mentioned before, this question was answered by Geil and Martin in [11]. Furthermore, in the general situation with d 1 ≤ · · · ≤ d m , this problem was solved in [1] in the case u 2 = −1 in order to determine the GHWs of the affine Cartesian codes. In order to proceed, we first introduce the following two notations:…”

“…We note that if u 2 = −1, then M r (u 1 , u 2 ) are simply the r-th generalized Hamming weights of AC q (u 1 , A). In the recent work [1], the generalized Hamming weights of affine Cartesian codes were completely determined. To answer the above question we introduce the following sets.…”

“…For a comprehensive reading on these notions, the reader is referred to [6]. Most of what follows in this section can be found in [1,Section 2]. We provide a somewhat detailed description of what will be used later for the sake of completeness and ease of readability.…”

“…The fundamental properties of affine Cartesian codes, such as their dimensions and the minimum distances, were obtained in [17]. Later in 2018, the generalized Hamming weights [1] of the affine Cartesian codes were completely determined. This generalizes the classical work [15] of Heijnen and Pelikaan towards the determination of all the generalized Hamming weights of the Reed-Muller codes.…”