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Unit $\mathbb{Z}_q$-Simplex codes of type α and zero divisor $\mathbb{Z}_q$-Simplex codes | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 4، دوره 8، شماره 2، شهریور 2023، صفحه 327-348 اصل مقاله (492.45 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2022.27363.1247 | ||
| نویسندگان | ||
| J. Mahalakshmi* ؛ J. Prabu؛ S. Santhakumar | ||
| Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India | ||
| چکیده | ||
| In this paper, we have punctured unit $\mathbb{Z}_q$-Simplex code and constructed a new code called unit $\mathbb{Z}_q$-Simplex code of type $\alpha$. In particular, we find the parameters of these codes and have proved that it is an $\left[\phi(q)+2, ~\hspace{2pt} 2, ~\hspace{2pt} \phi(q)+2 - \frac{\phi(q)}{\phi(p)}\right]$ $\mathbb{Z}_q$-linear code $\text{if} ~ k=2$ and $\left[\frac{\phi(q)^k-1}{\phi(q)-1}+\phi(q)^{k-2}, ~k,~ \frac{\phi(q)^k-1} {\phi(q)-1}+\phi(q)^{k-2}-\left(\frac{\phi(q)}{\phi(p)}\right)\left(\frac{\phi(q)^{k-1}-1}{\phi(q)-1}+\phi(q)^{k- 3}\right)\right]$ $\mathbb{Z}_q$-linear code if $k \geq 3, $ where $p$ is the smallest prime divisor of $q.$ For $q$ is a prime power and rank $k=3,$ we have given the weight distribution of unit $\mathbb{Z}_q$-Simplex codes of type $\alpha$. Also, we have introduced some new code from $\mathbb{Z}_q$-Simplex code called zero divisor $\mathbb{Z}_q$-Simplex code and proved that it is an $\left[ \frac{\rho^k-1}{\rho-1}, \hspace{2pt} k, \hspace{2pt} \frac{\rho^k-1}{\rho-1}-\left(\frac{\rho^{(k-1)}-1}{\rho-1}\right)\left(\frac{q}{p}\right) \right]$ $\mathbb{Z}_{q}$-linear code, where $\rho = q-\phi(q)$ and $p$ is the smallest prime divisor of $q.$ Further, we obtain weight distribution of zero divisor $\mathbb{Z}_q$-Simplex code for rank $k=3$ and $q$ is a prime power. | ||
| کلیدواژه‌ها | ||
| Unit Zq-Simplex codes of type α؛ Unit Zq-MacDonald code؛ Zero divisor Zq-Simplex code and Weight distribution | ||
| مراجع | ||
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