**Title:** Bounds on the covering radius of some classes code, simplex code and MacDonald code in *R*

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-21-00073; **Volume 4 / Issue 2 / Year 2022**, Pages 37-44

**Document Type:** Research Paper

**Author(s):** Panchanathan Chella Pandian ^{a}

^{a}Department of Mathematics, Srimad Andavan Arts and Science College(A), Tiruchirappalli-620005, Tamil Nadu, India

Received: 3 December 2021, Accepted: 2 June 2022, Published: 14 July 2022.

**Corresponding Author:** Panchanathan Chella Pandian (Email address: chellapandianpc@gmail.com; chella@andavancollege.ac.in)

**Full Text:** PDF

**Abstract**

In this paper, study of the codes in where , Its related parameter of codes over with different distance are discussed. The block repetition codes over are defined and the covering radius for block repetition codes, simplex code and macdonald code of type and type in are obtained.

**Keywords:** Code, finite ring, additive codes, parameter, different distance, block repetition code, simplex code, Macdonald code

**References:**

- T. Abualrub, I. Siap and N. Aydin,
*-additive cyclic codes*, IEEE Trans. Inform. Theory**60 (3)**, 1508–1514, 2014. - T. Aoki, P. Gaborit, M. Harada, M. Ozeki and P. Sole,
*On the covering radius of -codes and their lattices*, IEEE Trans. Inform. Theory**45 (6)**, 2162–2168, 1999. - C. Bachoc,
*Application of coding theory to the construction of modular lattices*, J. Combin. Theory Ser. A**78 (1)**, 92–119, 1997. - M. C. Bhandari, M. K. Gupta and A. K. Lal,
*On -simplex codes and their gray images*, In: Lecture Notes in Computer Science, 1719, Applied Algebra, Algebraic Algorithms and Error Correcting Codes, AAECC-13, 170–180, 1999. - M. Bilal, J. Borges, S. T. Dougherty and C. Fernandez-Cordoba,
*Maximum distance separable codes over and*, Des. Codes Cryptogr.**61 (1)**, 31–40, 2011. - J. Borges, S. T. Dougherty and C. Fernandez-Cordoba,
*Characterization and constructions of self-dual codes over*, Adv. Math. Commun.**6 (3)**, 287–303, 2012. - J. Borges, C. Fernandez-Cordoba, J. Pujol, J. Rifa and M. Villanueva,
*-linear codes: Generator matrices and duality*, Des. Codes Cryptogr.**54 (2)**, 167–179, 2010. - P. Chella Pandian,
*On covering radius of codes over where*, Discrete Math. Algorithms Appl.*u*^{2}= 0, using chinese euclidean distance**9 (2)**, 2017; Article ID: 1750017. - P. Chella Pandian,
*On covering radius of codes over where*, International Journal of Mathematics And its Applications,*u*^{2}= 0, using Bchoc distance**5 (4C)**, 277–282, 2017. - P. Chella Pandian and C. Durairajan,
*On the covering radius of some codes over where*International Journal of Research in Applied, Natural and Social Sciences, Impact Journal*u*^{2}= 0,**2 (1)**, 61–70, 2014. - P. Chella Pandian and C. Durairajan,
*On various parameters of -simplex codes for an even integer*, Algebra Discrete Math.*q***19 (2)**, 243–253, 2015. - P. Chella Pandian and C. Durairajan,
*On -linear and -simplex codes and its related parameters for*, J. Discrete Math. Sci. Cryptogr.*q*is a prime power**18 (1-2)**, 81–94, 2015. - P. Chella Pandian and J. Mahalakshmi,
*On -linear codes and its covering radius of some classes of linear codes in*, Adv. Math., Sci. J.**9 (3)**, 1165–1183, 2020. - G. D. Cohen, M. G. Karpovsky, H. F. Mattson and J. R. Schatz,
*Covering radius-survey and recent results*, IEEE Trans. Inform. Theory**31 (3)**, 328–343, 1985. - C. J. Colbourn and M. K. Gupta,
*On quaternary MacDonald codes*, Proc. Information Technology: Coding and Computing (ITCC) 212–215, 2003. - P. Delsarte,
*An algebraic approach to the association schemes of coding theory*, Philips Res. Repts Suppl., Netherlands, 1973. - P. Delsarte and V. Levenshtein,
*Association schemes and coding theory*, IEEE Trans. Inform. Theory**44 (6)**, 2477–2504, 1998. - S. Dodunekov and J. Simonis,
*Codes and projective multisets*. Electron. J. Comb.**5**, 1–23, 1998. - C. Durairajan,
*On covering codes and covering radius of some optimal codes*, Ph.D. Thesis, Department of Mathematics, IIT Kanpur, 1996. - M. K. Gupta,
*On some linear codes over ℤ*, Ph.D. Thesis, IIT Kanpur, 1999._{2s} - M. K. Gupta and C. Durairajan,
*On the covering radius of some modular codes*, Adv. Math. Commun.**8 (2)**, 129–137, 2014. - M. K. Gupta, D. G. Glynn and T. A. Gulliver,
*On senary simplex codes*, In: International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes (Ed. by S. Boztaş, I. E. Shparlinski), Lecture Notes in Computer Science,**2227**, Springer, Berlin, Heidelberg, 112–121, 2001.