Abstract
The explicit expressions for the 2n + 1 primitive idempotents in\(R_{p^ - } = F[x]/< x^{p^ - } - 1 > \), whereF is the field of prime power orderq and the multiplicative order ofq modulop n is ϕ(p n)/2 (n ≥ 1 andp is an odd prime), are obtained. An algorithm for computing the generating polynomials of the minimal QR cyclic codes of lengthp n, generated by these primitive idempotents, is given and hence some bounds on the minimum distance of some QR codes of prime length overGF(q)(q = 2, 3, ...) are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
David M. Burton,Elementary Number Theory, Wm.C. Brown Publishers, 1939.
Robert Calderbank and David B. Wales,Multiplying Vectors in binary quadratic residue Codes, SIAM J. Algebraic Discrete Methods, Vol. 3 (1982), no. 1, 43–55.
Neal H. McCoy,The Theory of Numbers, The McMillan Company London, 1965.
F.J. McWilliams and N.J.A. Sloane,The Theory of Error Correcting Codes, North Holland, Amsterdam, 1977.
Vera Pless,Introduction of the Theory of Error Correcting Codes, A Wiley-Interscience Publication, New York, 1981.
J.C.C.M. Ramijn and de Vroedt,The minimum distance of the [38, 19] ternary extended QR code is 11, IEEE Transactions on Information Theory, Vol. 30 (1984) No. 2(2) 405–407.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Batra, S., Arora, S.K. Minimal quadratic residue cyclic codes of lengthp n(p odd prime). Korean J. Comput. & Appl. Math. 8, 531–547 (2001). https://doi.org/10.1007/BF02941985
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02941985