Minimal quadratic residue cyclic codes of length 2 n



LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 n (n>1) be ϕ(2 n )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m−3, then the ring
$$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$
has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 n generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n−1 primitive idempotents ofR 2 n are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n−1 idempotents are also obtained. The casen=2, 3 is dealt separately.

AMS Mathematics Subject Classification

94B15 16S34 20C05 

Keywords and phrases

Cyclotomic cosets minimal cyclic codes quadratic residue codes generating polynomials primitive idempotents 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. K. Arora, Sudhir Batra, Stephen D. Cohen and Manju Pruthi,Primitive Idempotents of a cyclic Group algebra, Southeast Asian Bulletin of Mathematics26(4) (2002), 549–557.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S. K. Arora, Sudhir Batra and Stephen D. Cohen,Primitive Idempotents of a cyclic Group algebra II, Southeast Asian Bulletin of Mathematics (accepted for publication)Google Scholar
  3. 3.
    Sudhir Batra, Ph. D. Thesis, submitted to M. D. University Rohtak, India (1999).Google Scholar
  4. 4.
    Sudhir Batra and S. K. Arora,Minimal Quadratic Residue Cyclic Codes of Length p n, J. Appl. Math. & Computing(old:KJCAM)8(3) (2001), 531–547.MATHMathSciNetGoogle Scholar
  5. 5.
    Harriet Griffin,Elementary Theory of Numbers, McGraw-Hill Book Company, New York, 1954.MATHGoogle Scholar
  6. 6.
    F. J. McWilliams and N. J. A. Sloane,The Theory of Error Correcting Codes, North Holland, Amsterdam, 1977.Google Scholar
  7. 7.
    Vera Pless,Introduction of the Theory of Error Correcting Codes, A Wiley-Interscience Publication, New York, 1981.Google Scholar
  8. 8.
    Manju Pruthi and S. K. Arora,Minimal Codes of Prime-Power Length, Finite Fields and their Applications3 (1997), 99–113.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Paulo Ribenboin,Algebraic Numbers, Wiley-Interscience Publishers, New York, 1972.Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2005

Authors and Affiliations

  1. 1.Department of MathematicsT. I. T. & SBhiwaniIndia
  2. 2.Department of MathematicsMaharshi Dayanand UniversityRohtakIndia

Personalised recommendations