Well, one needs to prove then that such an a^m+b^m is never a primitive element (unless one takes this for granted given the previous answers, but I don't know if that was the idea).

True. I expected that the database containes all the designs, and testing resolubility seems at first glance something quick for these particular parameters. But yes, I would rather test the article than the database. I will continue searching.

Thanks. In the article (bottom of p. 326) it is said that there exists a (18,6,5) code by applying a result in R. C. Mullin, Resolvable designs and geometroids, Utilitas Math. 5 (1974), 137-149 (mathscinet link: ams.org/mathscinet-getitem?mr=0345841), unfortunately I could not access the article in order to try to construct a solution. But yes, it seems that one exists. How does this fit with the negative result in my database search? Is it possible that the definitions used are not exactly the same?

@Douglas Zare: sorry to surprise you :) , but I got the results file for that query, and there are no resolvable ones there: it does contain 582 lines that say "resolvable" : false and 0 that say true.

Thanks, we already tried to "find" our problem in nassrat.cs.dal.ca/ddb2 but the property of splitting in 3 groups of 6 seemed to be an extra condition not considered there. I have tried to understand the specifications of the DESIGN package (designtheory.org/software/gap_design/htm/CHAP002.htm), there is something there about "resolvable" block designs, which seems to fit.