We consider the geometric problem of determining the maximum number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n_q(r,h,f;s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>f</mi> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(h-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-spaces in the projective space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{PG}(r-1,q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>PG</mtext> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that each subspace of codimension f contains at most s elements. In terms of coding theory, this corresponds to additive codes with a large fth generalized Hamming weight. We also consider the dual problem. Here, we determine the minimum number <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$b_q(r,h,f;s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>f</mi> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(h-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-spaces in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{PG}(r-1,q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>PG</mtext> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that each subspace of codimension f contains at least s elements. We fully determine <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$b_2(5,2,2;s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as a function of s. We additionally give bounds and constructions for other parameters. For the computational result we partially use extensive integer linear programming computations.

Generalized Hamming weights of additive codes and geometric counterparts

期刊论文
发表日期：

2026年6月18日

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Jozefien D’haeseleer,
Sascha Kurz

摘要

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