Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle}
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913132266389504 |
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| author | Chahal, Seema Antil, Seema Maheshwary, Sugandha Khan, Manju |
| author_facet | Chahal, Seema Antil, Seema Maheshwary, Sugandha Khan, Manju |
| contents | Let $\mathbb{F}_{p^m}$ be the field containing $p^m$ elements where $p$ is an odd prime and $m \in \mathbb{N}$.
In this article, we propose a unified approach to the study of skew constacyclic codes of length $np^s$ over the ring $R_k = \mathbb{F}_{p^m}[u]/\langle u^k \rangle,$ where $n, s, k \in \mathbb{N}$ and $\gcd(n, p)=1$.
Consider the skew polynomial ring $R_k[x;Θ]$, where
$Θ$ is an automorphism of $R_k$ such that $xa = Θ(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{np^s} - λ$ of degree $l$ and multiplicity $j$ in $R_k[x;Θ]$, where $λ$ is an invertible element in $R_k$. In this article, we study skew constacyclic codes of length \(np^s\) over \(R_k\), which reduces to the study of skew polycyclic codes of length $jl$ associated with a polynomial \(f(x)^j\).
Using the fact that skew polycyclic codes associated with a polynomial \(f(x)^j\) can be described by the left ideal structure of the quotient ring $R_k[x;Θ]/\langle f(x)^{j}\rangle$, we investigate this class of codes for specific choices of $Θ$. In particular, if $λ$ is an invertible element of $\mathbb{F}_{p^m}$, we classify all left ideals and establish an isomorphism between skew cyclic and skew constacyclic codes, under suitable conditions.
Furthermore, we provide a comprehensive analysis of skew constacyclic codes of length $3p^s$ over $R_k$. Finally, we examine skew cyclic and skew negacyclic codes of length $6p^s$ over $R_k$ using the factorization of $x^{6p^s} - 1$ and $x^{6p^s} + 1$, respectively; with a complete case-by-case analysis. Examples demonstrating codes with optimal parameters are also included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15925 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle} Chahal, Seema Antil, Seema Maheshwary, Sugandha Khan, Manju Information Theory Rings and Algebras Let $\mathbb{F}_{p^m}$ be the field containing $p^m$ elements where $p$ is an odd prime and $m \in \mathbb{N}$. In this article, we propose a unified approach to the study of skew constacyclic codes of length $np^s$ over the ring $R_k = \mathbb{F}_{p^m}[u]/\langle u^k \rangle,$ where $n, s, k \in \mathbb{N}$ and $\gcd(n, p)=1$. Consider the skew polynomial ring $R_k[x;Θ]$, where $Θ$ is an automorphism of $R_k$ such that $xa = Θ(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{np^s} - λ$ of degree $l$ and multiplicity $j$ in $R_k[x;Θ]$, where $λ$ is an invertible element in $R_k$. In this article, we study skew constacyclic codes of length \(np^s\) over \(R_k\), which reduces to the study of skew polycyclic codes of length $jl$ associated with a polynomial \(f(x)^j\). Using the fact that skew polycyclic codes associated with a polynomial \(f(x)^j\) can be described by the left ideal structure of the quotient ring $R_k[x;Θ]/\langle f(x)^{j}\rangle$, we investigate this class of codes for specific choices of $Θ$. In particular, if $λ$ is an invertible element of $\mathbb{F}_{p^m}$, we classify all left ideals and establish an isomorphism between skew cyclic and skew constacyclic codes, under suitable conditions. Furthermore, we provide a comprehensive analysis of skew constacyclic codes of length $3p^s$ over $R_k$. Finally, we examine skew cyclic and skew negacyclic codes of length $6p^s$ over $R_k$ using the factorization of $x^{6p^s} - 1$ and $x^{6p^s} + 1$, respectively; with a complete case-by-case analysis. Examples demonstrating codes with optimal parameters are also included. |
| title | Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle} |
| topic | Information Theory Rings and Algebras |
| url | https://arxiv.org/abs/2605.15925 |