We present a new construction for snake-in-the-box codes and circuit codes. Our technique is based on arranging necklaces, i.e., equivalence classes of words under cyclic shifting, to form codes and gives 21 new codes of lengths 8-17, all having more words than the best previously known codes. Included among these are two snake-in-the-box codes: one of length 8 with 96 words, the other of length 10 with 340 words. We also report 15 new codes with small parameters. We prove the optimality of two of these codes and an already-known code. These last results were obtained using straightforward computer search.