We consider the problem of finding the minimal length <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n_q(k,d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a linear code over <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> of fixed dimension k and fixed minimum distance d. For ternary codes of dimension 6 this problem is solved for all but 70 values of d. In this paper, we resolve three of the undecided cases: <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d=344$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>344</mn> </mrow> </math></EquationSource> </InlineEquation>, 345, 346. The problem is tackled by associating with the linear codes in question certain minihypers with bounded point multiplicity. In this paper we make use of the characterization of the minihypers with parameters (66, 21), (67, 21) and (68, 21) in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${{\,\textrm{PG}\,}}(4,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>PG</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to rule out the existence of the minihypers with parameters (210, 68), (209, 68) and (207, 67) in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${{\,\textrm{PG}\,}}(5,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>PG</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This violates the existence of the hypothetical ternary codes with parameters [518, 6, 344], [519, 6, 345], [521, 6, 346], and implies the three exact values; <Equation ID="Equ9"> <EquationSource Format="TEX">$\begin{aligned}n_3(6,344)=519, n_3(6,345)=520, n_3(6,346)=522.\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>n</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>344</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>519</mn> <mo>,</mo> <msub> <mi>n</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>345</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>520</mn> <mo>,</mo> <msub> <mi>n</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>346</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>522</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>The proof is based on geometric arguments and is entirely computer-free.

Characterization of some minihypers in ${{\,\textrm{PG}\,}}(r,3)$ and the nonexistence of some ternary Griesmer codes

期刊论文
发表日期：

2026年1月5日

查看全文

Ivan Landjev,
Emiliyan Rogachev,
Assia Rousseva

摘要

We consider the problem of finding the minimal length $n_q(k,d)$ $n_{q} (k, d)$ of a linear code over $\mathbb {F}_q$ $F_{q}$ of fixed dimension k and fixed minimum distance d. For ternary codes of dimension 6 this problem is solved for all but 70 values of d. In this paper, we resolve three of the undecided cases: $d=344$ $d = 344$ , 345, 346. The problem is tackled by associating with the linear codes in question certain minihypers with bounded point multiplicity. In this paper we make use of the characterization of the minihypers with parameters (66, 21), (67, 21) and (68, 21) in ${{\,\textrm{PG}\,}}(4,3)$ $PG (4, 3)$ to rule out the existence of the minihypers with parameters (210, 68), (209, 68) and (207, 67) in ${{\,\textrm{PG}\,}}(5,3)$ $PG (5, 3)$ . This violates the existence of the hypothetical ternary codes with parameters [518, 6, 344], [519, 6, 345], [521, 6, 346], and implies the three exact values; $\begin{aligned}n_3(6,344)=519, n_3(6,345)=520, n_3(6,346)=522.\end{aligned}$ $\begin{matrix} n_{3} (6, 344) = 519, n_{3} (6, 345) = 520, n_{3} (6, 346) = 522 . \end{matrix}$ The proof is based on geometric arguments and is entirely computer-free.

Characterization of some minihypers in \({{\,\textrm{PG}\,}}(r,3)\) and the nonexistence of some ternary Griesmer codes

摘要