Weight distributions of some irreducible cyclic codes of length n

Let $$n=p_1^{\alpha _1}p_2^{\alpha _2}...p_r^{\alpha _r}$$ n = p 1 α 1 p 2 α 2 . . . p r α r , where $$p_i$$ p i ’s are distinct odd primes, be a positive integer and $$F_l$$ F l be a finite field of prime order l where $$gcd(l,n)=1$$ g c d ( l , n ) = 1 . Here, we compute the weight distributions of all irreducible cyclic codes of length n over $$F_l$$ F l for the case when multiplicative order of l modulo $$p_i^{\alpha _i}$$ p i α i ; $$O_{p_i^{\alpha _i}}(l):=2p_i^{\alpha _{i}-1}$$ O p i α i ( l ) : = 2 p i α i - 1 for each $$\alpha _i \ge 1$$ α i ≥ 1 . It is also shown that computation of weight distributions of irreducible cyclic codes of length $${p_1p_2...p_r}$$ p 1 p 2 . . . p r over $$F_l$$ F l is sufficient to compute the weight distributions of all irreducible cyclic codes of length n.