Generalized Hamming weights of additive codes and geometric counterparts
Jozefien D'haeseleer, Sascha Kurz
TL;DR
The paper connects finite projective geometry with additive coding theory by studying n_q(r,h,f;s) and its dual b_q(r,h,f;s), parameters that govern how many small-dimension subspaces can be arranged under codimension-f constraints. It translates generalized Hamming weights of additive codes into geometric packing problems in PG(r-1,q), and develops both asymptotic bounds and constructive methods (including LMRD-based schemes and Solomon–Stiffler-type partitions) to bound or realize these quantities. A central achievement is the complete determination of b_2(5,2,2;s) as a function of s, complemented by detailed bounds and constructions for other parameter regimes and a thorough analysis of blocking sets in PG(4,q). The work highlights the interplay between blocking sets, partial line spreads, and constant-dimension codes, offering new directions for exact results and open problems in higher-dimensional parameter regimes.
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.