A class of 1-generator quasi-cyclic codes

If R = F_q[x┐]/(x^m - 1), S = F_qn[x]/(x^m - 1), we define the mapping a_(x) → A(x) =σ₀^n-1a_i(x)α_i from Rⁿ onto S, where (α₀, α_i,..., α_n-1) is a basis for F_qn over F_q. This carries the q-ray 1-generator quasicyclic (QC) code R a_(x) onto the code RA(x) in S whose parity-check polynomial (p.c.p.) is defined as the monic polynomial h(x) over F_q of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of x_m 1 over F_q and F_qn are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal in S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for F_qn over F_q, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes.