A class of Zr-simplex codes and its gray images
*J. MahalakshmiCorresponding authorj_mahalakshmi@cb.amrita.eduDepartment of MathematicsAmrita School of Physical SciencesAmrita Vishwa VidyapeethamCoimbatore, Tamil Nadu, IndiaView full profile → , J. Prabuj_prabu11um47@gmail.comDepartment of MathematicsPSG Institute of Technology and Applied ResearchNeelambur, Coimbatore, Tamil Nadu, 641062, IndiaView full profile → , M. S. Sooraj Subrahmaniyansoorajms910@gmail.comDepartment of MathematicsAmrita School of Physical SciencesAmrita Vishwa VidyapeethamCoimbatore, Tamil Nadu, IndiaView full profile → , K. M. Sanmayasanmayakaruvat@gmail.comDepartment of MathematicsAmrita School of Physical SciencesAmrita Vishwa VidyapeethamCoimbatore, Tamil Nadu, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 May 2025
- Published Online:
- 03 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2624
- Pages:
- 1–16
Abstract
The various parameters of Zr-Simplex code under the Homogeneous and Lee weight metric for some particular cases of r with rank k are considered. Some fundamental properties like weight distributions of Zr-Simplex code are determined under the same metric. It is demonstrated that the Gray map generates several intriguing p-ary nonlinear codes of Zr-Simplex codes.
Keywords
Subject Classifications
References
[1] M. C. Bhandari, M. K. Gupta, and A. K. Lal, “On Z4 Simplex codes and their Gray images,” in Applied Algebra, Algebraic Algorithms and Error Correcting Codes, vol. 1719, pp. 170-180, Springer (1999).
[2] E. Byrne, M. Greferath, A. Kohnert, and V. Skachek, “New bounds for codes over finite Frobenius rings,” Des., Codes, Cryptogr., vol. 42, no. 3, pp. 289-301 (2010).
[3] C. Carlet, “Z2k-linear codes,” IEEE Trans. Inf. Theory, vol. 44, no. 4, pp. 1543-1547 (1998).
[4] I. Constantinescu and W. Heise, “A metric for codes over residue class rings of integers,” Probl. Pered. Inform., vol. 33, no. 3, pp. 22-28 (1997).
[5] P. C. Pandian and C. Durairajan, “On Zq-linear and Zq-Simplex codes and their related parameters for q a prime power,” J. Discrete Math. Sci. Cryptogr., vol. 18, no. 1-2, pp. 81-94 (2015).
[6] C. Durairajan and J. Mahalakshmi, “On codes over integers modulo q,” Adv. Appl. Discrete Math., vol. 15, no. 2, pp. 125-143 (2015).
[7] C. Durairajan, J. Mahalakshmi, and P. C. Pandian, “On the Zq-Simplex codes and its weight distribution for dimension 2,” Discrete Math. Algorithms Appl., vol. 7, no. 3, p. 1550030 (2015).
[8] M. K. Gupta and C. Durairajan, “On the covering radius of some modular codes,” Adv. Math. Commun., vol. 8, no. 2, pp. 129-137 (2014).
[9] A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z4-linearity of Kerdock, Preparata, Goethals, and related codes,” IEEE Trans. Inf. Theory, vol. 40, no. 2, pp. 301-319 (1994).
[10] M. Kiermaier and J. Zwanzger, “New ring-linear codes from dualization in projective Hjelmslev geometries,” Des., Codes, Cryptogr., vol. 66, no. 1-3, pp. 39-55 (2013).
[11] J. Prabu, J. Mahalakshmi, and S. Santhakumar, “Unit Zr-Simplex codes of type α and zero divisor Zr-Simplex codes,” Commun. Comb. Optim., vol. 8, no. 2, pp. 327-348 (2023).
[12] J. Prabu, J. Mahalakshmi, C. Durairajan, and S. Santhakumar, “On some punctured codes of Zq-Simplex codes,” Discrete Math. Algorithms Appl., pp. 2250012 (2021).
[13] B. Yildiz and Z. Ödemis Özger, “A generalization of the Lee weight to Zpk,” TWMS J. Appl. Eng. Math., vol. 2, no. 2, pp. 145-153 (2012).
[14] M. Pavone, “On the weight distribution of perfect binary codes,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 26, no. 1, pp. 271-279 (2023).
[15] S. Ling and J. T. Blackford, “Zpk-Linear codes,” IEEE Trans. Inf. Theory, vol. 48, no. 9, pp. 2592-2605 (2002).
[16] M. Greferath and S. E. Schmidt, “Gray isometries for finite chain rings and a nonlinear ternary (36, 312, 15) code,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2522-2524 (1999).
