To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$, minimum distance $d$ with $(d + 1) \log_4 n + O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth's balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing.