...

  • Home
  • Articles
    • Current Issue
    • Archives
  • Authors
    • Author Guidelines
    • Policies
    • Downloads
  • Editors
  • Reviewers
...

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 5 | Year 2022 | Article Id. IJMTT-V68I5P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I5P507

Generating, Minimal Polynomial and Minimum Distance of Quadratic Residue Codes (QR-Codes) of Length 4p^n


Ranbir Singh, Manoj Kumar, Vinod Bhatia
Received Revised Accepted Published
16 Mar 2022 11 May 2022 17 May 2022 07 Jun 2022
Abstract

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.

Keywords
Generating polynomials, Minimal polynomial and Minimum distance.
References

[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).

Citation :

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

  • PDF
  • Abstract
  • Keywords
  • References
  • Citation
Abstract Keywords References Citation
  • Home
  • Authors Guidelines
  • Paper Submission
  • APC
  • Archives
  • Downloads
  • Open Access
  • Publication Ethics
  • Copyrights Infringement
  • Journals
  • FAQ
  • Contact Us

Follow Us

Copyright © 2025 Seventh Sense Research Group® . All Rights Reserved