Necklaces, self-reciprocal polynomials, and \(q\)-cycles (Q471794)

Summary: Let \(n\ge 2\) be a positive integer and \(q\) a prime power. Consider necklaces consisting of \(n\) beads, each of which has one of the given \(q\) colors. A primitive \(C_n\)-orbit is an equivalence class of \(n\) necklaces closed under rotation. A \(C_n\)-orbit is self-complementary when it is closed under an assigned color matching. In the work of \textit{R. L. Miller} [Discrete Math. 22, 25--33 (1978; Zbl 0395.05004)], it is shown that there is a 1-1 correspondence between the set of primitive, self-complementary \(C_n\)-orbits and that of self-reciprocal irreducible monic (srim) polynomials of degree \(n\). Let \(N\) be a positive integer relatively prime to \(q\). A \(q\)-cycle \(\bmod N\) is a finite sequence of nonnegative integers closed under multiplication by \(q\). In the work of \textit{Z.-X. Wan} [Lectures on finite fields and Galois rings. River Edge, NJ: World Scientific (2003; Zbl 1028.11072)], it is shown that \(q\)-cycles \(\bmod N\) are closely related to monic irreducible divisors of \(x^N-1\in \mathbb F_q[x]\). Here, we show that: (1) \(q\)-cycles can be used to obtain information about srim polynomials; (2) there are correspondences among certain \(q\)-cycles and \(C_n\)-orbits; (3) there are alternative proofs of Miller's results in the work of Miller (1978, loc. cit.) based on the use of \(q\)-cycles.