Constructions of perfect Mendelsohn designs

Holey Steiner pentagon systems

“…Below are the required pentagons. n = 5: (0, 1, 13, 9, 2) mod 15, (0, 6, 12, 3, 9), (1,7,13,4,10), (2,8,14,5,11). Proof.…”

“…More specifically, we shall make use of the following two constructions. For similar applications of this technique, the reader is referred to [4,5,6,15,16,18] In the construction of GDDs or PBDs, the technique of ''filling in holes'' plays an important role. This simple technique also works effectively for the construction of a variety of other combinatorial structures (see, for example, [5]).…”

“…(1, 2, 5, 4, 12), (1,3,12,9,10), (1,4,3,6,8), (1,5,11,10,6), (1,9,2,8,11), (2,3,8,7,12), (2,4,10,3,11), (2, 7, 9, 5, 10), (3,7,6,11,9), (4,6,12,10,7), (4,8,5,7,11), (5,6,9,8,12). …”

“…It is known [14] that the spectrum of SPSs is precisely the set of all n ≡ 1 or 5 (mod 10), except n = 15, for which no such system exists. SPSs are also known to be useful in the construction of other types of designs such as perfect Mendelsohn designs with block-size five (see [4]). …”

Authentication perpendicular arrays APA1(2, 7, ?)

“…This construction is inspired by some known constructions for Steiner pentagon systems ( [11]), perfect Mendelsohn designs ( [1]) and BIB designs ( [20]). Let k be an odd integer.…”