Primitive Idempotents and Minimum Distance Bounds for Some Cyclic Codes of Length 8pn

Let F(= GF(q)) is a finite field of order q and G be a cyclic group of order n. For n ≥ 1, the cyclic codes of length 8pn over F can be viewed as ideals in the semi-simple ring R8pn ≡ GF(q)[x] <x8pn−1> [13]. The primitive idempotents of minimal cyclic codes of length m in case, when order of q modulo m is ϕ(m) for m=2,4,pn,2pn were computed in [8, 11]. The primitive idempotents of length pn with order of q modulo pn is ϕ(pn) 2 were obtained in [12] and minimal quadratic residue codes of length pn in [9]. Cyclic codes of length 2pn over F, where order of q modulo 2pn is ϕ(2pn) 2 were discussed in [10]. Minimal cyclic codes of length pnq, where p and q are distinct odd primes were derived in [1, 3]. Further, when order of q modulo pn is ϕ(pn), the minimal cyclic codes of length 8pn were discussed in [4, 5]. The minimal cyclic codes of length 8pn, where q ≡ 1(mod 8) or q ≡ 5(mod 8) and order of q modulo pn is ϕ(pn) 2 were obtained in [6, 7]. Irreducible cyclic codes of length 4pn and 8pn, where q ≡ 3(mod 8) and p/(q −1) were obtained in [2]. In present paper, we obtained cyclic codes of length 8pn over F where q is of the form 8k + 3 or 8k + 7 and order of q modulo pn is ϕ(pn) 2 . The q-cyclotomic cosets modulo 8pn are obtained in section 2 and corresponding primitive idempotents in section 3. In section 4, we discussed generating polynomials and dimensions for the corresponding cyclic codes of length 8pn. The minimum distance or the bounds for minimum distance of these codes are obtained in section 5. At the end, an example is discussed to illustrate the various parameters for these codes