R. Julian R. Abel

Some progress on ( v , 4, 1) difference families and optical orthogonal codes

Some new classes of optimal (v, 4, 1) optical orthogonal codes are constructed. First, mainly by using perfect difference families, we establish that such an optimal OOC exists for v ≤ 408, v ≠ 25. We then look at larger (p, 4, 1) OOCs with p prime; some of these codes have the nice property that the missing differences are the (r - 1)th roots of unity in Zp, (r being the remainder of the Euclidean division of p by 12) and we prove that for r = 5 or 7 they give rise to (rp, 4,1) difference families. In this way we are able to give a strong indication about the existence of (5p,4,1) and (7p,4, 1) difference families with p a prime ≡ 5, 7 mod 12 respectively. In particular, we prove that for a given prime p ≡ 7 mod 12, the existence of a (7p,4, 1) difference family is assured (1) if p < 10,000 or (2) if ω is a given primitive root unity in Zp and we have 3 ≡ ωi (mod p) with gcd(i, p-1/6) < 20.Finally, we remove all undecided values of v ≤ 601 for which a cyclic (v, 4, 1) difference family exists, and we give a few cyclic pairwise balanced designs with minimum block size 4.

Some difference matrix constructions and an almost completion for the existence of triplewhist tournaments TWh( v )

A necessary condition for the existence of a triplewhist tournament TWh(v) is v ≡ 0 or 1 (mod 4); this condition is known to be sufficient except for v = 5, 9, 12, 13 and possibly v = 17, 57, 65, 69, 77, 85, 93, 117, 129, 153. In this paper, we remove all the possible exceptions except v = 17. This provides an almost complete solution for the more than 100 year old problem on the existence of triplewhist tournaments TWh(v). By applying frame constructions and product constructions, several new infinite classes of Z-cyclic triplewhist tournaments are also obtained. A couple of new cyclic difference matrices are also obtained.

Modified group divisible designs with block size 5 and λ = 1

In this paper we investigate the existence of 5-MGDDs of typemn. Some new incomplete transversal designs are also given.

The Existence of Four HMOLS with Equal Sized Holes

In this paper, we present several new constructions for k holey mutually orthogonal Latin squares (HMOLS) of type gn. We concentrate mainly on k=4; here, for all but two values of n, namely 6 and 15, only a finite number of unsolved cases remain. Some new sets of 5 and 6 HMOLS are also given, in particular 5 HMOLS(2q) for q≥63 or q an odd prime power between 6 and 62, plus 6 HMOLS(4q)for q an odd prime power between 8 and 60.

Balanced Incomplete Block Designs with Block Size 7

The object of this paper is the construction of balanced incomplete block designs with k=7. This paper continues the work begun by Hanani, who solved the construction problem for designs with a block size of 7, and with λ=6, 7, 21 and 42. The construction problem is solved here for designs with λ > 2 except for v=253, λ= 4,5 ; also for λ= 2, the number of unconstructed designs is reduced to 9 (1 nonexistent, 8 unknown).

The existence of three idempotent IMOLS

In this paper it is shown that an idempotent TD(5,m)- TD(5,n) exists whenever the known necessary condition m ≥ 4n+ 1 is satisfied, except when (m, n)=(6, 1) and possible when (m, n)= (10, 1). For m > 60 and n ≤ 10, we also indicate where several idempotent TD(k, m)-TD(k, n)s for k = 6, 7 can be found.

Resolvable Balanced Incomplete Block Designs with BlockSize 8

The necessary condition for the existence of a resolvable balanced incomplete block design on v points, with λ = 1 and k = 8, is that v ≡ 8 mod 56. With the exception of 66 values of v, this condition is shown to be sufficient. The largest exceptional value of v is 24480.

Resolvable balanced incomplete block designs with block size 5

Abstract A necessary condition for the existence of a resolvable balanced incomplete block design on v points with λ=2 and k=5, is that v≡5 mod 10 . This condition is shown to be sufficient for v>395, with at most 13 exceptions below this value. Also given are a new (185,5,1) RBIBD, (v,5,4) RBIBDs for v=110 and 140, and (45,5,λ) RBIBDs for λ⩾3.

Direct constructions for certain types of HMOLS

Abstract In this paper, we present 16 new types of three holey mutually orthogonal Latin squares (3-HMOLS for short) with equal-sized holes, along with 7 sets of 4-HMOLS, 2 sets of 6-HMOLS, and 34 sets of 7-HMOLS. We also provide some new sets of HMOLS that have n holes of size 2 and one hole of size 3.

Generalized whist tournament designs

In this study a new class of tournament designs is introduced. In particular, each game of the tournament involves several (two or more) teams competing against one another. The tournament is also required to satisfy certain balance conditions that are imposed on each pair of players. These balance conditions are related to both the total number of players on each team and the number of teams in each game. In one sense, these balance conditions represent a generalization of the balance requirements for whist tournaments although the games in a whist tournament involve, exclusively, two two-player teams. Several techniques for constructing these new tournament designs are developed and theorems guaranteeing infinite classes of such designs are proven.