Some cyclic codes of length 2p n

Abstract

Explicit expressions for 4n + 2 primitive idempotents in the semi-simple group ring R 2 p n ≡ G F(q) [x] <x 2 p n −1> are obtained, where p and q are distinct odd primes; n ≥ 1 is an integer and q has order φ(2 p n ) 2 modulo 2 p n . The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2 p n generated by these 4n + 2 primitive idempotents are discussed. For n = 1, the properties of some (2 p, p) cyclic codes, containing the above minimal cyclic codes are analyzed in particular. The minimum weight of some subset of each of these (2 p, p) codes are observed to satisfy a square root bound.

Minimal cyclic codes of length 2p^n

“…In this paper, we have extended the results of Batra, Arora [8]. We consider the case when k = 2p n , where p and l are distinct odd primes,…”

“…, the complete set of primitive idempotents in R k are obtained by Batra, Arora [8]. For k = p n q (n 1), p and q distinct odd primes where l is primitive root modulo p n and q both with gcd( ) ( ), 2 ( q p n   ) = 2, the primitive idempotents in R k are obtained by Bakshi and Raka [3].…”

Weight distributions of cyclic codes of length tlm

“…On the other hand, several papers investigated the primitive idempotent elements of F q [x]/(x n − 1) to determine the minimum Hamming weights of cyclic codes of length n = tl m in [1,3,4,8]. Sharma and Bakshi [21] gave the weight distributions of irreducible cyclic codes of length l m , where the multiplicative order of q modulo l m is one among φ(l m ), l i , and 2l i .…”

Irreducible cyclic codes of length 4pn and 8pn

“…Singh and Pruthi presented explicit expressions for all the 4t 1 t 2 + 2t 1 + 2t 2 + 1 primitive idempotents in the ring R l . (5) In [1], l = m t , t ≥ 1, where m is odd prime and ord m t (q) = φ(m t )/2. Arora et al…”

“…(6) In [5], l = 2m t , t ≥ 1, where m is odd prime and ord 2m t (q) = φ(2m t )/2. Batra and Arora got explicit expressions for 4t + 2 primitive idempotents in R l .…”