Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="254" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {R}_\lambda = \mathbb {F}_q + s_1\mathbb {F}_q + s_2\mathbb {F}_q + \ldots + s_\lambda \mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mi>λ</mi> </msub> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>s</mi> <mi>λ</mi> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$s_\rho ^2 = s_\rho$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>s</mi> <mi>ρ</mi> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>s</mi> <mi>ρ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$s_\rho s_\varphi = s_\varphi s_\rho = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mi>ρ</mi> </msub> <msub> <mi>s</mi> <mi>φ</mi> </msub> <mo>=</mo> <msub> <mi>s</mi> <mi>φ</mi> </msub> <msub> <mi>s</mi> <mi>ρ</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="126" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho , \varphi = 1, 2, \ldots , \lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>,</mo> <mi>φ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho \ne \varphi$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>≠</mo> <mi>φ</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$q = p^e$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>e</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where p is an odd prime and e is a positive integer. In this article, we have shown that if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$q> 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, then any linear code <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq9.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> is equivalent to a Euclidean linear complementary dual (LCD) code, and if <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$0 \le \ell \le e$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>ℓ</mi> <mo>≤</mo> <mi>e</mi> </mrow> </math></EquationSource> </InlineEquation>, then the code <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> is equivalent to an <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1641_Article_IEq12.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>-Galois LCD code.

A Study of Galois LCD Codes Over a Family of Non-chain Rings

期刊论文
发表日期：

2025年5月16日

查看全文

Rishi Raj,
Sachin Pathak,
Marbarisha M. Kharkongor,
Dipendu Maity

摘要

Let $\mathcal {R}_\lambda = \mathbb {F}_q + s_1\mathbb {F}_q + s_2\mathbb {F}_q + \ldots + s_\lambda \mathbb {F}_q$ $R_{λ} = F_{q} + s_{1} F_{q} + s_{2} F_{q} + \dots + s_{λ} F_{q}$ , where $s_\rho ^2 = s_\rho$ $s_{ρ}^{2} = s_{ρ}$ and $s_\rho s_\varphi = s_\varphi s_\rho = 0$ $s_{ρ} s_{φ} = s_{φ} s_{ρ} = 0$ for $\rho , \varphi = 1, 2, \ldots , \lambda$ $ρ, φ = 1, 2, \dots, λ$ with $\rho \ne \varphi$ $ρ \neq φ$ , and $q = p^e$ $q = p^{e}$ , where p is an odd prime and e is a positive integer. In this article, we have shown that if $q> 3$ $q > 3$ , then any linear code $\mathcal {C}$ $C$ over $\mathbb {F}_q$ $F_{q}$ is equivalent to a Euclidean linear complementary dual (LCD) code, and if $0 \le \ell \le e$ $0 \leq ℓ \leq e$ , then the code $\mathcal {C}$ $C$ is equivalent to an $\ell$ $ℓ$ -Galois LCD code.