AbstractIn this article, we first show that a group divisible 3‐design with block sizes from {4, 6}, index unity and group‐type 2m exists for every integer m≥ 4 with the exception of m = 5. Such group divisible 3‐designs play an important role in our subsequent complete solution to the existence problem for directed H‐designs DHλ(m, r, 4, 3)s. We also consider a way to construct optimal codes capable of correcting one deletion or insertion using the directed H‐designs. In this way, the optimal single‐deletion/insertion‐correcting codes of length 4 can be constructed for all even alphabet sizes. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 136–146, 2009