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

Document Type : Original paper


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


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.


Main Subjects

[1] R.C. Bose and R.C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the Macdonald codes, J. Combin. Theory Ser. A 1 (1966), no. 1, 96–104.
[2] Durairajan C. and J. Mahalakshmi, On codes over integers modulo q, Adv. Appl. Discrete Math. 15 (2015), no. 2, 125–143.
[3] K. Chatouh, K. Guenda, T.A. Gulliver, and L. Noui, Simplex and Macdonald codes over $R_q$, J. Appl. Math. Comput. 55 (2017), no. 1, 455–478.
[4] P. Chella Pandian and C. Durairajan, On $Z_q$-linear and $Z_q$-Simplex codes and its related parameters for $q$ is a prime power, J. Discrete Math. Sci. Cryptogr. 18 (2015), no. 1-2, 81–94.
[5] C.J. Colbourn and M.K. Gupta, On quaternary macdonald codes, Proceedings ITCC 2003. International Conference on Information Technology: Coding and Computing, IEEE, 2003, pp. 212–215.
[6] A. Dertli and Y. Cengellenmis, Macdonald codes over the ring $F_2 + vF_2$, Int. J. Algebra 5 (2011), no. 20, 985–991.
[7] C. Ding and C. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Math. 340 (2017), no. 10, 2415–2431.
[8] C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math. 313 (2013), no. 4, 434–446.
[9] C. Durairajan, J. Mahalakshmi, and P.C. Pandian, On the $Z_q$-Simplex codes and its weight distribution for dimension 2, Discrete Math. Algorithms Appl. 7 (2015), no. 0, 1550030.
[10] R.A. Fisher, A system of confounding for factors with more than two alternatives, giving completely orthogonal cubes and higher powers, Annals of Eugenics 12 (1943), no. 1, 283–290.
[11] R.A. Fisher, The theory of confounding in factorial experiments in relation to the theory of groups, Annals of Eugenics 11 (1941), no. 1, 341–353.
[12] M.K. Gupta, On some linear codes over Z2s, Ph.D. thesis, Indian Institute of Technology, Kanpur, 1999.
[13] M.K. Gupta, D.G. Glynn, and T.A. Gulliver, On senary simplex codes, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Springer, 2001, pp. 112–121.
[14] Y. Liu, C. Ding, and C. Tang, Shortened linear codes over finite fields, IEEE Transactions on Information Theory (2021).
[15] J.E. MacDonald, Design methods for maximum minimum-distance error-correcting codes, IBM Journal of Research and Development 4 (1960), no. 1, 43–57.
[16] J. Mahalakshmi and C. Durairajan, On the $Z_q$-Macdonald code and its weight distribution of dimension 3, J. Math. Comput. Sci. 6 (2016), no. 2, 173–187.
[17] A. Patel, Maximal $q$-nary linear codes with large minimum distance (Corresp.), IEEE Transactions on Information Theory 21 (1975), no. 1, 106–110.
[18] J. Prabu, J. Mahalakshmi, C. Durairajan, and S. Santhakumar, On some punctured codes of $Z_q$-Simplex codes, Discrete Math. Algorithms Appl. (2021), 2250012.
[19] Y. Wang and J. Gao, Macdonald codes over the ring $F_p + vF_p + v^2F_p$, Comput. Appl. Math. 38 (2019), no. 4, 1–15.
[20] A. Yardi and R. Pellikaan, On shortened and punctured cyclic codes, arXiv preprint arXiv:1705.09859 (2017).