A note on an upper bound of traceability codes

Blackburn, Etzion and Ng showed in a paper in 2010 that there exist 2-traceability codes of length l of size cqdl/4e, where the constant c depends only on l. The question remains as to what the best possible c may be. A well-known construction using error-correcting codes with high minimum distance gives 2-traceability codes of size cqdl/4e with c ≥ 1. However, in the same paper, an example of a 2-traceability code of length 3 with size 3 2 (q − 1) was given, which shows that c > 1 in some situations, and that there are traceability codes that are bigger than the construction using error-correcting codes. Here we give an upper bound 4q − 3 for 2-traceability codes of length 4 and give an example of (l− 1)traceability codes of length l with size l l−1 (q−1). This example also gives a 2-traceability code of length 4 larger than any codes constructed using the error-correcting code construction.