Relative generalized Hamming weights of affine Cartesian codes

Abstract

We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics. This generalizes the previous works on the determination of relative generalized Hamming weights of Reed-Muller codes by Geil and Martin, as well as the determination of all the generalized Hamming weights of the affine Cartesian codes by Beelen and Datta. The author is supported by a postdoctoral fellowship from DST-RCN grant INT/NOR/RCN/ICT/P-03/2018.

Relative generalized Hamming weights of evaluation codes

“…Proof. By Theorem 3.3, we have that The relative generalized Hamming weights of an affine Cartesian code with respect to a smaller affine Cartesian code have been computed by M. Datta [14] in the following way. Take the set…”

“…The footprint of S/I, denoted ∆ ≺ (I), is the set of all standard monomials of S/I. The footprint has been extensively utilized for the study of evaluation codes [14,17,18,20,21,25,26,28].…”

“…O. Geil and M. Stefano obtained the relative generalized Hamming weights of q-ary Reed-Muller codes using the footprint bound [16]. M. Datta followed the footprint bound technique and presented a combinatorial formula for the relative generalized Hamming weights of affine Cartesian codes [14]. These works introduced methods to study certain evaluation codes.…”