Volume 68 | Issue 5 | Year 2022 | Article Id. IJMTT-V68I5P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I5P507
Received | Revised | Accepted | Published |
---|---|---|---|
16 Mar 2022 | 11 May 2022 | 17 May 2022 | 07 Jun 2022 |
Generating, Minimal polynomial and Minimum distance of all 3(2n+1) quadratic residue codes(QR-Codes) of length 4𝑝 𝑛 over GF(l), where 𝑝, 𝑙 are distinct odd primes, does not divide 3 , are obtained.
[1] S.K. Arora and Manju Pruthi, Minimal Cyclic Codes Length 2pn, Finite Field and Their Applications, 5 (1999) 177-187.
[2] Gurmeet K. Bakshi and Madhu Raka, Minimal Cyclic Codes of Length Pnq, Finite Fields Appl. 9(4) (2003) 432-448.
[3] Sudhir Batra and S.K. Arora, Minimal Quadratic Residue Cyclic Codes of Length Pn(P Odd Prime), Korean J. Comput & Appl. Math. 8(3) (2001) 531-547.
[4] Sudhir Batra and S.K. Arora, Some Cyclic Codes of Length 2pn(P Odd Prime), Design Codes Cryptography , 57(3) (2010).
[5] F.J. Mac Williams & N.J.A. Sloane ; the Theory of Error Correcting Codes Bell Laboratories Murray Hill Nj 07974 U.S.A.
[6] Manju Pruthi and S.K. Arora, Minimal Cyclic Codes of Prime Power Length, Finite Field and Their Application, 3 (1997) 99-113.
[7] Raka, M., Bakshi ,G.K.; Sharma,A.,Dumir,V.C. Cyclotomic Numbers and Primitive Idempotents In the Ring GF(q)[x]/(xpn - 1), Finite Field & Their Appl.3, 2 (2004) 653-673.
[8] Ferraz,R.A.,Millies,C.P., Idempotents In Group Algebras and Minimal Abelian Codes, Finite Fields and Their Appl,13(2) (2007) 982-993.
[9] A.Sahni and P.T.Sehgal,Minimal Cyclic Codes of Length Pnq, Finite Fields Appl. 18 (2012) 1017-1036.
[10] Ranjeet Singh and Manju Pruthi, Primitive Idempotents of Quadratic Residue Codes of Length Pn Qm, Int.J.Algebra, 5(2011) 285- 294.
[11] S. Batra and S.K. Arora,Minimal Quadratic Residue Cyclic Codes of Length Pn (P Odd Prime),Korean J. Comput and Appl. Math, 8(3)(2001) 531- 547.
[12] S.Rani, I.J.Singh and S.K. Arora, Minimal Cyclic Codes of Length 2Pnq (P Odd Prime), Bull.Calcutta.Math Society ,106(4)(2014)281-296.
[13] S. Rani, P,Kumar and I.J.Singh, Minimal Cyclic Codes of Length 2Pn,Int. J.Algebra 7, 1- 4 (2013) 79 - 90.
[14] S. Rani, P,Kumar and I.J.Singh, Quadratic Residues Codes of Prime Power Length Over Z4,J.Indian Math.Soc.New Series ,78(1-4)(2011) 155 -161.
[15] S. Rani, I.J.Singh and S.K.Arora,Primitive Idempotents of Irre-Ducible Cyclic Code sof Length Pnqm ,Far East Journal of Math.Sciences, 77(1) (2013) 17 - 32.
[16] R. Singh, V. Kumar and I.J. Singh,`` Generalized Cyclotomic Cosets Modulo 4pn Aryabhatta J.of Maths & Info. 10(1) (2020) 93-96.
[17] Vanlint, J.H. Generalised Quadratic Residue Macwilliams, F.J. Codes, Ieee Trans. Infor. Theory, 24(6) (1978) 730- 737.
[18] Vanlint, J.H. Introduction to Coding Theory, Springer- Verlag, (1999).
[19] Vanlint, J.H. on the Minimum Distance of Wilson, Richard M. Cyclic Codes Ieee Trans. Infor. Theory, 32(1) (1986) 23-40.
[20] Vermani, L.R. Elements of Algebraic Coding Theory, Chapman & Hall.
[21] Ward, H.N. Quadratic Residue Codes and Symplectic Group, Journal of Algebra 29,150-171 (1974) 168- 173.
[22] Mc Coy, N.H. the Theory of Numbers, the Mcmillan Company, New York, (1971).
[23] Niven, I., Zuckerman, H.S. an Introduction To the Theory of Numbers, John Wiley & Sons, Inc, New York, (1960).
[24] Huffman, D.G. Coding Theory, Marcel Dekker, Inc. New York, (1991).
[25] Jenson, R.A. Information Sets As Permutation Cycles for Quadratic Residue Codes, Internat.J. Math. Sci, 5 (1982).
Ranbir Singh, Manoj Kumar, Vinod Bhatia, "Generating, Minimal Polynomial and Minimum Distance of Quadratic Residue Codes (QR-Codes) of Length 4p^n," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 5, pp. 40-43, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I5P507