In this paper, we study the b-symbol distance of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-constacyclic codes of length <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$p^\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>ℓ</mi> </msup> </math></EquationSource> </InlineEquation> over the ring <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq5.gif" Format="GIF" Height="31" Rendition="HTML" Resolution="72" Type="Linedraw" Width="238" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A} = \mathbb {F}_{p^m} + u \mathbb {F}_{p^m} + u^2 \mathbb {F}_{p^m} = \frac{\mathbb {F}_{p^m}[u]}{\langle u^3 \rangle }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>p</mi> <mi>m</mi> </msup> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>p</mi> <mi>m</mi> </msup> </msub> <mo>+</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>p</mi> <mi>m</mi> </msup> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>p</mi> <mi>m</mi> </msup> </msub> <mrow> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> </mrow> </mrow> <mrow> <mo stretchy="false">⟨</mo> <msup> <mi>u</mi> <mn>3</mn> </msup> <mo stretchy="false">⟩</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, where p is a prime number, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon \in \mathbb {F}_{p^m}^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <msup> <mi>p</mi> <mi>m</mi> </msup> </mrow> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> and m are positive integers, and b is an integer satisfying <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq8.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le b \le \left\lfloor \frac{p}{2} \right\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>b</mi> <mo>≤</mo> <mfenced close="⌋" open="⌊"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we classify all maximum distance separable (MDS) b-symbol <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-constacyclic codes of length <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$p^\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>ℓ</mi> </msup> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2423_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>.

On the MDS b-symbol of repeated-root constacyclic codes of prime power length over $ \mathbb {F}_{p^{m}} + u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}} $

期刊论文
发表日期：

2025年3月4日

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Youssef Ahendouz,
Ismail Akharraz

摘要

In this paper, we study the b-symbol distance of $\varepsilon $ $ε$ -constacyclic codes of length $p^\ell $ $p^{ℓ}$ over the ring $\mathcal {A} = \mathbb {F}_{p^m} + u \mathbb {F}_{p^m} + u^2 \mathbb {F}_{p^m} = \frac{\mathbb {F}_{p^m}[u]}{\langle u^3 \rangle }$ $A = F_{p^{m}} + u F_{p^{m}} + u^{2} F_{p^{m}} = \frac{F_{p^{m}} [u]}{⟨ u^{3} ⟩}$ , where p is a prime number, $\varepsilon \in \mathbb {F}_{p^m}^*$ $ε \in F_{p^{m}}^{*}$ , $\ell $ $ℓ$ and m are positive integers, and b is an integer satisfying $1 \le b \le \left\lfloor \frac{p}{2} \right\rfloor $ $1 \leq b \leq ⌊\frac{p}{2}⌋$ . Additionally, we classify all maximum distance separable (MDS) b-symbol $\varepsilon $ $ε$ -constacyclic codes of length $p^\ell $ $p^{ℓ}$ over $\mathcal {A}$ $A$ .

On the MDS b-symbol of repeated-root constacyclic codes of prime power length over \( \mathbb {F}_{p^{m}} + u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}} \)

摘要