Weight Distribution of Minimal Cyclic Codes over a Finite Field
DOI:
https://doi.org/10.26438/ijcse/v11i6.4547Keywords:
Primitive root, Weight distribution, Minimal Cyclic CodesAbstract
Let Fq be the finite field with q elements, p, q be two odd primes with gcd(2p, q) = 1, multiplicative order of q modulo 2p^m is p^d (0?d?m-1), m ? 1 be an integer. In this paper, we obtain weight distribution of all the minimal(irreducible) cyclic codes of length 2pm over Fq by using their generating polynomials.
References
[1] A.Sharma, G. K. Bakshi and M. Raka,”The weight distribu- tion of some irreducible cyclic codes of length pn”, Finite Fields Appl.,Vol 18, Issue 1,pp.144-159, 2012.
[2] A.Sharma, G. K. Bakshi and M. Raka,”The weight distribution of irre- ducible cyclic codes of length 2n”, Finite Fields Appl.,Vol.13, Issue 4, pp.1086-1095, 2007.
[3] A.Sharma, G. K. Bakshi, V. C. Dumir and M. Raka,” Cyclotomic numbers and primitive idempotents in the ring (GF(q)[x])/((x^(p^n )-1)) “, Finite Fields Appl.,Vol.10, Issue 4, pp.653-673, 2004.
[4] C.Ding,”The weight distribution of some irreducible cyclic codes,”IEEE Trans.Inform.Theory ,Vol.55, Issue 3, pp.955-960, 2009.
[5] M. Pruthi and S. K. Arora(1997), “Minimal Codes of Prime-Power Length”, Finite Fields Appl., Vol.3, Issue 3, pp.99-113, 1997.
[6] S.Batra and S.K.Arora,”Some cyclic codes of length 2pn”,Codes Designs and Crypto., Vol.61, Issue 01, pp.41-69, 2011.
[7] S. K. Arora and M. Pruthi, “Minimal cyclic codes of length 2pn”, Finite Fields Appl., Vol.5, pp.177-187, 1999.
[8] R. Singh and M. Pruthi,”Primitive idempotents of quadratic residue codes of length pnqm,”Int.J.Algebra,Vol.5, pp.285-294, 2011.
[9] S. Batra and S.K. Arora,”Minimal quadratic residue cyclic codes of length pn(p odd prime),”Korean J. Comput and Appl. Math. , Vol.8, Issue 3, pp.531-547, 2001.
[10] S.Rani,I.J.Singh and S.K. Arora, ”Minimal cyclic codes of length 2pnq(p odd prime),”Bull.Calcutta.Math Society ,Vol.106, Issue 4, pp.281-96, 2014.
[11] S. Rani, P,Kumar and I.J.Singh, ”Minimal cyclic codes of length 2pn,”Int. J. Algebra, Vol.7, Issue 1-4, pp.79-90, 2013.
[12] S. Rani, P,Kumar and I.J.Singh, ”Quadratic residues codes of prime power length over Z4,”J.Indian Math.Soc.New Series , Vol.78, Issue 1-4, pp.155-161, 2011.
[13] Seema Rani, I.J.Singh and S.K.Arora,”Primitive idempotents of irre- ducible cyclic codes of length pnqm”,Far East Journal of Mathematical Sciences, Vol.77, Issue 1, pp.17-32, 2013.
[14] P. Kumar, M. Sangwan and S. K. Arora,” The weight distributions of some irreducible cyclic codes of lengthpn and 2pn,Adv. Math. Commun, Vol.9, pp.277-289, 2015.
[15] Riddhi, K. Singh and P. Kumar,” Weight distributions of some irre- ducible cyclic codes of length n, Indian Jour. Pure Appli. Math. Vol.53, pp.1073-1082, 2022.
[16] Inderjit Singh, Pankaj Kumar and Monika Sangwan,” Primitive idempotents in a semi- simple ring”,Asian E.Journal of mathematics,Vol.16, Issue 4, 2023.
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