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Let \(\mathbb {Z}_{2}=\{0,1\}\) , \(\mathfrak {R_{1}}=\mathbb {Z}_{2}+u\mathbb {Z}_{2}+u^{2}\mathbb {Z}_{2}+u^{3}\mathbb {Z}_{2}\) , where \(u^4=0\) and \(\mathfrak {R_{u^{k}}}=\mathbb {Z}_{2}+u\mathbb {Z}_{2}+\cdots +u^{k-1 }\mathbb {Z}_{2}\) , where \(u^{k}=0\) and k4. In this article, we study \(\mathbb {Z}_{2}\mathfrak {R_{1}}\mathfrak {R_{u^k}}\) -additive cyclic codes and their structural properties. The additive cyclic codes are characterized as \(\mathfrak {R_{u^k}}[y]\) -submodules of the ring \(\mathcal {S}_{\beta _{1},\beta _{2}, \beta _{3}}={\mathbb {Z}_{2}[y]/\langle y^{\beta _{1}}-1\rangle } \times {\mathfrak {R_{1}}[y]/\langle y^{\beta _{2}}-1\rangle }\times {\mathfrak {R_{u^k}}[y]/\langle y^{\beta _{3}}-1\rangle }.\) We obtain the minimal generating polynomials and smallest spanning sets of the above-specified codes. Furthermore, we determine the generating set for \(\mathbb {Z}_{2}\mathfrak {R_{1}}\mathfrak {R_{u^k}}\) -additive cyclic codes of length \((\beta _{1}, \beta _{2}, \beta _{3})\) through the factorization of \(y^{\beta _{2}}-1\) and \(y^{\beta _{3}}-1\) into pairwise coprime monic polynomials, where \(\beta _{2}\) and \(\beta _{3}\) are odd positive integers.

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On Construction of Additive Cyclic Codes over \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u^4]\mathbb {Z}_{2}[u^k]\)

  • 会议论文

  • 发表日期：

    2025年

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  • Hai Q. Dinh,
  • Mohd Asim,
  • Ghulam Mohammad,
  • Washiqur Rehman

摘要

Let \(\mathbb {Z}_{2}=\{0,1\}\) , \(\mathfrak {R_{1}}=\mathbb {Z}_{2}+u\mathbb {Z}_{2}+u^{2}\mathbb {Z}_{2}+u^{3}\mathbb {Z}_{2}\) , where \(u^4=0\) and \(\mathfrak {R_{u^{k}}}=\mathbb {Z}_{2}+u\mathbb {Z}_{2}+\cdots +u^{k-1 }\mathbb {Z}_{2}\) , where \(u^{k}=0\) and k4. In this article, we study \(\mathbb {Z}_{2}\mathfrak {R_{1}}\mathfrak {R_{u^k}}\) -additive cyclic codes and their structural properties. The additive cyclic codes are characterized as \(\mathfrak {R_{u^k}}[y]\) -submodules of the ring \(\mathcal {S}_{\beta _{1},\beta _{2}, \beta _{3}}={\mathbb {Z}_{2}[y]/\langle y^{\beta _{1}}-1\rangle } \times {\mathfrak {R_{1}}[y]/\langle y^{\beta _{2}}-1\rangle }\times {\mathfrak {R_{u^k}}[y]/\langle y^{\beta _{3}}-1\rangle }.\) We obtain the minimal generating polynomials and smallest spanning sets of the above-specified codes. Furthermore, we determine the generating set for \(\mathbb {Z}_{2}\mathfrak {R_{1}}\mathfrak {R_{u^k}}\) -additive cyclic codes of length \((\beta _{1}, \beta _{2}, \beta _{3})\) through the factorization of \(y^{\beta _{2}}-1\) and \(y^{\beta _{3}}-1\) into pairwise coprime monic polynomials, where \(\beta _{2}\) and \(\beta _{3}\) are odd positive integers.