Existence of T*(3, 4,v)‐codes

A word of length k over an alphabet Q of size v is a vector of length k with coordinates taken from Q. Let Q*4 be the set of all words of length 4 over Q. A T*(3, 4, v)-code over Q is a subset C*⊆ Q*4 such that every word of length 3 over Q occurs as a subword in exactly one word of C*. Levenshtein has proved that a T*(3, 4, vv)-code exists for all even v. In this paper, the notion of a generalized candelabra t-system is introduced and used to show that a T*(3, 4, v)-code exists for all odd v. Combining this with Levenshtein's result, the existence problem for a T*(3,4, v)-code is solved completely. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 42–53, 2005.