Let p be a prime, and let <InlineEquation ID="IEq4"> <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> denote the finite field of order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$q = p^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$m \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Firstly, we introduce and analyze the algebraic structures of skew polycyclic codes over the ring <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R = \mathbb {F}_q + v\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mi>v</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$v^2 = v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>. Then, we investigate skew polycyclic codes of length <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha +\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {F}_q R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation> as a generalization of skew polycyclic codes over R. Subsequently, we study the algebraic structures of these codes by determining their generator polynomials and minimal generating sets. Furthermore, we establish the structure of the dual codes corresponding to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$(\theta , \Theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-cyclic codes, which constitute a special class of skew polycyclic codes. As an application of our study, several optimal linear codes over <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> are provided.

Skew polycyclic codes over $\mathbb {F}_{q}R$

期刊论文
发表日期：

2026年7月3日

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Juan Li,
Yunbo Tian,
Fanghui Ma

摘要

Let p be a prime, and let $\mathbb {F}_q$ $F_{q}$ denote the finite field of order $q = p^m$ $q = p^{m}$ where $m \ge 2$ $m \geq 2$ . Firstly, we introduce and analyze the algebraic structures of skew polycyclic codes over the ring $R = \mathbb {F}_q + v\mathbb {F}_q$ $R = F_{q} + v F_{q}$ , where $v^2 = v$ $v^{2} = v$ . Then, we investigate skew polycyclic codes of length $\alpha +\beta $ $α + β$ over $\mathbb {F}_q R$ $F_{q} R$ as a generalization of skew polycyclic codes over R. Subsequently, we study the algebraic structures of these codes by determining their generator polynomials and minimal generating sets. Furthermore, we establish the structure of the dual codes corresponding to $(\theta , \Theta )$ $(θ, Θ)$ -cyclic codes, which constitute a special class of skew polycyclic codes. As an application of our study, several optimal linear codes over $\mathbb {F}_{q}$ $F_{q}$ are provided.

Skew polycyclic codes over \(\mathbb {F}_{q}R\)

摘要