Generalized Hamming weights of additive codes and geometric counterparts
This work addresses a combinatorial geometry problem with connections to coding theory, but the results are incremental and limited to specific parameters.
The paper studies the geometric problem of determining the maximum number of (h-1)-spaces in projective space such that each subspace of codimension f contains at most s elements, which corresponds to additive codes with large generalized Hamming weights. It fully determines b_2(5,2,2;s) as a function of s and provides bounds and constructions for other parameters.
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.