Cartesian square-free codes

TL;DR

The paper introduces Cartesian square-free codes, defined by evaluating square-free monomials over Cartesian sets, and studies their generalized Hamming weights using the footprint bound from commutative algebra. It provides sharp bounds and explicit formulas for $d_r$ in favorable regimes, translating these results to projective-space evaluation codes. The authors establish how to compute weight hierarchies via shadow sizes, present exact expressions under size constraints, and connect affine and projective evaluation codes through tensor-product and puncturing relations, enabling explicit GHWs for a broad family of codes with potential applications in coding theory and related areas.

Abstract

The generalized Hamming weights (GHWs) of a linear code C extend the concept of minimum distance, which is the minimum cardinality of the support of all one-dimensional subspaces of C, to the minimum cardinality of the support of all r-dimensional subspaces of the code. In this work, we introduce Cartesian square-free codes, which are linear codes generated by evaluating square-free monomials over a Cartesian set. We use commutative algebraic tools, specifically the footprint bound, to provide explicit formulas for some of the GHWs of this family of codes, and we show how we can translate these results to evaluation codes over the projective space.

Theorems & Definitions (21)

• Definition 2.1
• Theorem 2.2: Monotonicity
• Definition 2.3
• Theorem 2.4: Duality
• Theorem 3.1
• Theorem 3.2
• proof
• Remark 3.3
• Lemma 4.1
• proof