Generalized Hamming weights of additive codes and geometric counterparts

TL;DR

This paper explores geometric configurations in projective spaces related to generalized Hamming weights of additive codes, providing exact results, bounds, and constructions using computational methods.

Contribution

It introduces new bounds and exact values for the maximum and minimum configurations of subspaces in projective spaces linked to additive code properties.

Findings

Determined $b_2(5,2,2;s)$ exactly as a function of $s$.

Provided bounds and constructions for various parameters.

Used integer linear programming for computational results.

Abstract

We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding theory terms we are dealing with additive codes that have a large $f$th generalized Hamming weight. We also consider the dual problem of the minimum number $b_q(r,h,f;s)$ of $(h-1)$-spaces in $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ contains at least $s$ elements. We fully determine $b_2(5,2,2;s)$ as a function of $s$. We additionally give bounds and constructions for other parameters. For the computational results we partially use extensive integer linear programming computations.