Traceability schemes are used to protect the rights of intellectual-property owners against illegal re-distribution. In order to trace a pirate copy back to a user who has been involved in making and re-distributing the pirate copy, a traceability scheme requires a code which contains a sufficient number of codewords and a traitor-tracing algorithm. Codes with the identifiable parent property (IPP codes) and traceability codes (TA codes) have been extensively studied. IPP codes include all TA codes as special instances. However, TA codes usually implement efficient tracing algorithms, while IPP codes do not always have efficient tracing algorithms. A major theoretical challenge is to derive more codes which have efficient tracing algorithms. The contributions of this paper include a new class of traceability codes, as well as an efficient tracing algorithm for the new traceability codes. The proposed tracing algorithm has the same complexity as the traditional tracing algorithm and can outperform the traditional tracing algorithm.