Cartesian square-free codes

TL;DR

This paper introduces Cartesian square-free codes, a new class of linear codes generated by evaluating square-free monomials over Cartesian sets, and provides explicit formulas for their generalized Hamming weights using algebraic tools.

Contribution

It defines Cartesian square-free codes and derives explicit formulas for some of their GHWs using commutative algebraic methods, extending to evaluation codes over projective space.

Findings

Explicit formulas for GHWs of Cartesian square-free codes

Application of footprint bound in deriving GHWs

Extension of results to projective evaluation codes

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.