Generalized Hamming weights of additive codes and geometric counterparts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914519178018816 |
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| author | D'haeseleer, Jozefien Kurz, Sascha |
| author_facet | D'haeseleer, Jozefien Kurz, Sascha |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Hamming weights of additive codes and geometric counterparts D'haeseleer, Jozefien Kurz, Sascha Combinatorics Information Theory 94Bxx, 51E22 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. |
| title | Generalized Hamming weights of additive codes and geometric counterparts |
| topic | Combinatorics Information Theory 94Bxx, 51E22 |
| url | https://arxiv.org/abs/2512.16327 |