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 c q^l/4 where the constant c dependsonly on l. The question remains as to what the best possible c may be. Awell-known construction using error-correcting codes with high minimumdistance gives 2-traceability codes of size c q^l/4 with c ≥ 1. However, inthe same paper, an example of a 2-traceability code of length 3 withsize 3/2 (q − 1) was given, which shows that c > 1 in some situations, andthat there are traceability codes that are bigger than the constructionusing 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)-traceabilitycodes 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 theerror-correcting code construction.