Unit $\mathbb{Z}_q$-Simplex codes of type α and zero divisor $\mathbb{Z}_q$-Simplex codes

Document Type : Original paper

Authors

Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India

Abstract

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.

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References

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Communications in Combinatorics and Optimization
Volume 8, Issue 2
June 2023
Pages 327-348

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