<p>In this paper, we study <i>k</i>-dimensional double-twisted Reed-Solomon (DTRS) codes over finite fields <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_822_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <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>, having twists <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_822_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{t} = (1, 3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">t</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and general hooks <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_822_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{h} = (h_0, h_1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">h</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_822_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="149" /> </InlineMediaObject> <EquationSource Format="TEX">$0 \le h_0 \le h_{1} \le k-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>≤</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>≤</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We give necessary and sufficient conditions for such DTRS codes to be AMDS and AAMDS. We also enumerate AMDS and AAMDS DTRS codes for certain values of their parameters. Further, we show that the dimensions of the hull of these DTRS codes are small and vary from 0 to 4 with certain parameters. We also give some examples to illustrate the results of the paper.</p>

Double-twisted Reed-Solomon codes with twists (1, 3) and general hooks over finite fields

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发表日期：

2025年7月9日

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Kapish Chand Meena,
Maheshanand Bhaintwal,
Ambrish Awasthi,
Rajendra Kumar Sharma

摘要

In this paper, we study k-dimensional double-twisted Reed-Solomon (DTRS) codes over finite fields $\mathbb {F}_q$ $F_{q}$ , having twists $\varvec{t} = (1, 3)$ $t = (1, 3)$ and general hooks $\varvec{h} = (h_0, h_1)$ $h = (h_{0}, h_{1})$ , where $0 \le h_0 \le h_{1} \le k-1$ $0 \leq h_{0} \leq h_{1} \leq k - 1$ . We give necessary and sufficient conditions for such DTRS codes to be AMDS and AAMDS. We also enumerate AMDS and AAMDS DTRS codes for certain values of their parameters. Further, we show that the dimensions of the hull of these DTRS codes are small and vary from 0 to 4 with certain parameters. We also give some examples to illustrate the results of the paper.