Title: Primitive idempotents and generator polynomials of some minimal cyclic codes of length pⁿq^m

Authors: Pankaj Kumar; S.K. Arora; Sudhir Batra

Addresses: Department of Mathematics, Guru Jambheshwar University, Hisar 125001, India ' Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India ' Department of Mathematics, DCR University of Science and Technology, Murthal, Sonepat 131039, India

Abstract: Let p, q and l be distinct odd primes such that l is a primitive root modulo pⁿ as well as modulo q^m with g.c.d. (φ(p),φ(q)) = 2. Then the explicit expressions for the complete set of 2mn + m + n + 1 primitive idempotents of the minimal cyclic codes of length pⁿq^m over GF(l) are obtained. An algorithm is also given to factorise the polynomial (xⁿ - 1) over GF(k), where n is an integer such that g.c.d. (n, k) = 1. Using the algorithm generator polynomials of the above minimal cyclic codes can be computed numerically. Some bounds on the minimum distance of these minimal cyclic codes are also obtained.

Keywords: cyclotomic cosets; primitive idempotents; generator polynomials; cyclic codes; minimum distance.

DOI: 10.1504/IJICOT.2014.066087

International Journal of Information and Coding Theory, 2014 Vol.2 No.4, pp.191 - 217

Accepted: 22 Mar 2014
Published online: 02 Dec 2014 *

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