Unit Zq-Simplex codes of type α and zero divisor Zq-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 Zq-Simplex code  and constructed a new code called unit Zq-Simplex code of type α. In particular, we find the parameters of  these codes and have proved that it is an [ϕ(q)+2, 2, ϕ(q)+2−ϕ(q)ϕ(p)] Zq-linear code if k=2 and [ϕ(q)k−1ϕ(q)−1+ϕ(q)k−2, k, ϕ(q)k−1ϕ(q)−1+ϕ(q)k−2−(ϕ(q)ϕ(p))(ϕ(q)k−1−1ϕ(q)−1+ϕ(q)k−3)] Zq-linear code if k≥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 Zq-Simplex codes  of type α. Also, we have introduced some new code from  Zq-Simplex code called zero divisor Zq-Simplex code and proved that it is an [ρk−1ρ−1,k,ρk−1ρ−1−(ρ(k−1)−1ρ−1)(qp)] Zq-linear code, where ρ=q−ϕ(q) and p is the smallest prime divisor of q. Further, we obtain  weight distribution of  zero divisor Zq-Simplex code for rank k=3 and q is a prime power.

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References

[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 Rq, J. Appl. Math. Comput. 55 (2017), no. 1, 455–478.
[4] P. Chella Pandian and C. Durairajan, On Zq-linear and Zq-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 F2+vF2, 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 Zq-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 Zq-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 Zq-Simplex codes, Discrete Math. Algorithms Appl. (2021), 2250012.
[19] Y. Wang and J. Gao, Macdonald codes over the ring Fp+vFp+v2Fp, 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).
Communications in Combinatorics and Optimization
Volume 8, Issue 2
June 2023
Pages 327-348

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