An Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0001 design is a triple urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0002, where urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0003 is a set of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0004 points, urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0005 a partition of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0006 into urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0007 disjoint sets of size urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0008, and urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0009 a set of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0010‐element transverses of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0011, such that each urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0012‐element transverse of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0013 is contained in exactly one of them. In 1990, Mills determined the existence of an Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0014 design with urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0015. In this paper, an efficient construction shows that an Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0016 exists for any integer urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0017 with urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0018. Consequently, the necessary and sufficient conditions for the existence of an Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0019 design are urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0020, urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0021, and urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0022, with a definite exception urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0023.