Finally, we study the structure of self-dual $\mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$.
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Key Findings & Data:Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $\mathbb{Z}_{2}[u^r,u^s]$-linear codes.
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Conclusion & Impact:Finally, we study the structure of self-dual $\mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.
In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $\mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $\mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $\mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.
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