A class of Unit $\mathbb Z_{r}$-Simplex codes

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, Amrita School of Physical Sciences Coimbatore, Amrita Vishwa Vidyapeetham, India.

2 Department of Mathematics, PSG Institute of Technology and Applied Research, Neelambur, Coimbatore, Tamilnadu - 641 062, India.

3 Department of Mathematics, Amrita School of Physical Sciences Coimbatore, Amrita Vishwa Vidyapeetham, India

Abstract

In this paper, we determined the parameters of the unit $\mathbb Z_r$-Simplex code under the Homogeneous weight metric and showed that it is an $\left[ \frac{\rho^{k-1}}{\rho-1}, ~k~, \rho^{k-1}((p-1)p^{m-2})\right]$ $Z_r$-Simplex code if $m=2$, $\left[ \frac{\rho^{k-1}}{\rho-1}, ~k~, \rho^{k-2}((p-2)p^{2m-2}+2p^{m-1})\right]$ if $m>2$, where $r=p^m$ with rank $k$. Further, we obtain the weight distribution of the $\mathbb Z_r$-Simplex code under the Homogeneous weight metric for the particular rank $k=2$.

Keywords

Main Subjects

References

[1] T. Aoki, P. Gaborit, M. Harada, M. Ozeki, and P. Solé, On the covering radius of $Z_4$ codes and their lattices, IEEE Trans. Inform. Theory 45 (1999), no. 6, 2162–2168. https://doi.org/10.1109/18.782168
[2] M.C. Bhandari, M.K. Gupta, and A.K. Lal, On $Z_4$-simplex codes and their gray images, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Springer, 1999, pp. 170–180.
[3] A. Bonnecaze, P. Solé, C. Bachoc, and B. Mourrain, Type II codes over Z4, IEEE Trans. Inform. Theory 43, 969–976.
https://doi.org/10.1109/ISIT.1997.613114
[4] C. Carlet, $Z_{2^k}$-linear codes, IEEE Trans. Inform. Theory 44 (1998), 1543–1547.
[5] I. Constantinescu and W. Heise, A metric for codes over residue class rings of integers, Probl. Pered. Inform. 33 (1997), no. 3, 22–28.
[6] J.H. Conway and N.J.A. Sloane, Self-dual codes over the integers modulo 4, J. Comb. Theory Ser. A 62 (1993), no. 1, 30–45. https://doi.org/10.1016/0097-3165(93)90070-O
[7] C. Ding, C. Li, N. Li, and Z. Zhou, Three-weight cyclic codes and their weight distributions, Discrete Math. 339 (2016), no. 2, 415–427. https://doi.org/10.1016/j.disc.2015.09.001
[8] S.T. Dougherty and J.L. Kim, Constructions of self-dual codes over chain rings, Int. J. Inf. Coding Theory 1 (2010), no. 2, 171–190.
[9] S.T Dougherty, B. Yildiz, and S. Karadeniz, Cyclic codes over Rk Gray maps and their binary images, Finite Fields Appl. 17 (2011), no. 3, 205–219.
[10] C. Durairajan and J. Mahalakshmi, On codes over integers modulo q, Advances Appl. Discrete Math. 15 (2015), no. 2, 125–143. https://doi.org/10.17654/AADMApr2015_125_143
[11] C. Durairajan, J. Mahalakshmi, and P. Chella Pandian, On the $Z_q$-simplex codes and its weight distribution for dimension 2, Discrete Math Algorithms Appl. 7 (2015), no. 3, 1550030. https://doi.org/10.1142/S1793830915500305
[12] H. Gluesing-Luerssen, On the weight distribution of convolutional codes, Linear Algebra Appl. 408 (2005), 298–326.
https://doi.org/10.1016/j.laa.2005.06.023
[13] M. Greferath and S.E. Schmidt, Gray isometries for finite chain rings and a nonlinear ternary (36, 312, 15), IEEE Trans. Inform. Theory 45 (1999), no. 7, 2522–2524. https://doi.org/10.1109/18.796395
[14] M.K. Gupta and C. Durairajan, On the covering radius of some modular codes, Adv. Math. Comm. 8 (2014), no. 2, 129–137.
[15] M.K. Gupta, D.G. Glynn, and T. Aaron Gulliver, On senary simplex codes, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Springer, 2001, pp. 112–121.
[16] R.W. Hamming, Error detecting and error correcting codes, Bell System Technical J. 29 (1950), no. 2, 147–160.
https://doi.org/10.1002/j.1538-7305.1950.tb00463.x
[17] A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, The$ _Z$4-linearity of her dock, preparata, goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301–319. https://doi.org/10.1109/18.312154
[18] T. Honold and A.A. Nechaev, Weight functions on finite rings and their modules, Finite Fields and Their Applications 5 (1999), no. 4, 424–452.
[19] M. Karpovsky, On the weight distribution of binary linear codes, IEEE Trans. Inf. Theory 25 (2003), no. 1, 105–109.
https://doi.org/10.1109/TIT.1979.1056001
[20] M. Kiermaier and J. Zwanzger, New ring-linear codes from dualization in projective Hjelmslev geometries, Des. Codes, Cryptogr. 66 (2013), no. 1, 39–55. https://doi.org/10.1007/s10623-012-9650-1
[21] S. Ling and J.T. Blackford, $Z_{p^k}$-linear codes, IEEE Trans. Inform. Theory 48 (2002), no. 9, 2592–2605.
[22] C. Ma, L. Zeng, Y. Liu, D. Feng, and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory 57 (2011), no. 1, 397–402. https://doi.org/10.1109/TIT.2010.2090272
[23] J. Mahalakshmi, J. Prabu, and S. Santhakumar, Unit $Z_q$-simplex codes of type α and zero divisor $Z_q$-simplex codes, Commun. Comb. Optim. 8 (2023), no. 2, 327–348. https://doi.org/10.22049/cco.2022.27363.1247
[24] S. Minjia, T. Honold, P. Solé, Y. Qiu, R. Wu, and Z. Sepasdar, The geometry of two-weight codes over Zpm, IEEE Trans. Inf. Theory 67 (2021), no. 12, 7769–7781. https://doi.org/10.1109/TIT.2021.3114636
[25] A.A. Nechaev and T. Khonol’d, Weighted modules and representations of codes, Problemy Peredachi Informatsii 35 (1999), no. 3, 18–39.
[26] M. Özen and V. Şiap, On the existence of perfect linear codes over $Z_4$ with respect to homogeneous weight, Appl. Math. Sci. 6 (2012), no. 41, 2005 – 2011.
[27] P.C. 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. https://doi.org/10.1080/09720529.2014.903699
[28] J. Prabu, J. Mahalakshmi, C. Durairajan, and S. Santhakumar, On some punctured codes of $Z_q$-simplex codes, Discrete Math Algorithms Appl. 14 (2022), no. 6, 2250012. https://doi.org/10.1142/S1793830922500124
[29] Y. Song and J. Yang, Weight distribution of two classes of linear codes with a  few weights, Sci. China Inf. Sci. 63 (2020), no. 7, 179103. https://doi.org/10.1007/s11432-018-9610-9
[30] X. Wang, D. Zheng, and H. Liu, Several classes of linear codes and their weight distributions, Appl. Algebra Eng. Commun. Comput. 30 (2019), no. 1, 75–92. https://doi.org/10.1007/s00200-018-0359-x
[31] J. Yang, M. Xiong, C. Ding, and J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inf. Theory 59 (2013), no. 9, 5985–5993. https://doi.org/10.1109/TIT.2013.2266731
[32] B. Yildiz, A combinatorial construction of the Gray map over Galois rings, Discrete Math. 309 (2009), no. 10, 3408–3412. https://doi.org/10.1016/j.disc.2008.09.002
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 18 April 2026

Files

Share

How to cite

Statistics