An <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-linear code of minimum distance d is called complete if it is not contained in a larger <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-linear code of minimum distance d. In this paper, we classify <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-linear complete symmetric rank-distance codes in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M_{3\times 3}(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> up to equivalence. This includes the classification of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-linear maximum symmetric rank-distance codes in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M_{3\times 3}(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our approach is mainly geometric, and our results contribute towards the classification of nets of conics in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{PG}(2, q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>PG</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Linear complete symmetric rank-distance codes

期刊论文
发表日期：

2026年1月26日

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Nour Alnajjarine,
Michel Lavrauw

摘要

An $\mathbb {F}_q$ $F_{q}$ -linear code of minimum distance d is called complete if it is not contained in a larger $\mathbb {F}_q$ $F_{q}$ -linear code of minimum distance d. In this paper, we classify $\mathbb {F}_q$ $F_{q}$ -linear complete symmetric rank-distance codes in $M_{3\times 3}(\mathbb {F}_q)$ $M_{3 \times 3} (F_{q})$ up to equivalence. This includes the classification of $\mathbb {F}_q$ $F_{q}$ -linear maximum symmetric rank-distance codes in $M_{3\times 3}(\mathbb {F}_q)$ $M_{3 \times 3} (F_{q})$ . Our approach is mainly geometric, and our results contribute towards the classification of nets of conics in $\textrm{PG}(2, q)$ $PG (2, q)$ .