*i*. Then

Let there is an . For or , has been determined by Hanani, and for or , has been determined by the first author. In this paper, we investigate the case . A necessary condition for is . It is known that , and that there is an for all with a possible exception . We need to consider the case . It is proved that there is an for all with an exception and a possible exception , thereby, .

The first paper in this series initiated a study of Sylow theory for quasigroups and Latin squares based on orbits of the left multiplication group. The current paper is based on so-called pseudo-orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo-orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well behaved in the newly identified class of conflatable quasigroups, which provides a unified treatment of Moufang, Bol, and conjugacy closure properties. Relationships between sectional and Lagrangean properties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo-orbits. An upper bound is given on the size of a pseudo-orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invariants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15.

It is shown that, if is a nontrivial 2- symmetric design, with , admitting a flag-transitive automorphism group *G* of affine type, then , *p* an odd prime, and *G* is a point-primitive, block-primitive subgroup of . Moreover, acts flag-transitively, point-primitively on , and is isomorphic to the development of a difference set whose parameters and structure are also provided.

An *Euler tour* of a hypergraph (also called a *rank-2 universal cycle* or *1-overlap cycle* in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, using a graph-theoretic approach, we prove that every triple system with at least two triples is eulerian, that is, it admits an Euler tour. Horan and Hurlbert have previously shown that for every admissible order >3, there exists a Steiner triple system with an Euler tour, while Dewar and Stevens have proved that every cyclic Steiner triple system of order >3 and every cyclic twofold triple system admits an Euler tour.

External difference families (EDFs) are a type of combinatorial designs that originated from cryptography. Many combinatorial objects are closely related to EDFs, such as difference sets, difference families, almost difference sets, and difference systems of sets. Constructing EDFs is thus of significance in theory and practice. In this paper, earlier ideas of constructing EDFs proposed by Chang and Ding (2006), and Huang and Wu (2009), are further explored. Consequently, new infinite classes of EDFs are obtained and some previously known results are extended.

The problem of classifying cyclic Steiner quadruple systems (CSQSs) is considered. A computational approach shows that the number of isomorphism classes of such designs with orders 26 and 28 is 52,170 and 1,028,387, respectively. It is further shown that CSQSs of order 2*p*, where *p* is a prime, are isomorphic iff they are multiplier equivalent. Moreover, no CSQSs of order less than or equal to 38 are isomorphic but not multiplier equivalent.

In this note, we show that for each Latin square *L* of order , there exists a Latin square of order *n* such that *L* and differ in at most cells. Equivalently, each Latin square of order *n* contains a Latin trade of size at most . We also show that the size of the smallest defining set in a Latin square is .

The purpose of this paper is to classify all pairs , where is a nontrivial 2- design, and acts transitively on the set of blocks of and primitively on the set of points of with sporadic socle. We prove that there exists only one such pair : is the unique 2-(176,8,2) design and , the Higman–Sims simple group.

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result that shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order is smaller than (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most . In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3*q*, then it consists of the union of a blocking set and a covering set apart from a few points and lines.

The existence problem of a -cycle frame of type is now solved for any quadruple .

A tight Heffter array is an matrix with nonzero entries from such that (i) the sum of the elements in each row and each column is 0, and (ii) no element from appears twice. We prove that exist if and only if both *m* and *n* are at least 3. If *H* has the property that all entries are integers of magnitude at most , every row and column sum is 0 over the integers, and *H* also satisfies ), we call *H* an integer Heffter array. We show integer Heffter arrays exist if and only if . Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both are even.

Multi-layered Youden rectangles are introduced. These new designs include double and triple Youden rectangles as subdesigns. Though many double Youden rectangles are known, the triples, introduced in 1994, have to now yielded few nontrivial examples. Two infinite series of multi-layered Youden rectangles are constructed, and so also many new triple Youden rectangles.

*Paratopism* is a well-known action of the wreath product on Latin squares of order *n*. A paratopism that maps a Latin square to itself is an *autoparatopism* of that Latin square. Let Par(*n*) denote the set of paratopisms that are an autoparatopism of at least one Latin square of order *n*. We prove a number of general properties of autoparatopisms. Applying these results, we determine Par(*n*) for . We also study the proportion of all paratopisms that are in Par(*n*) as .

In this paper, we show that partial geometric designs can be constructed from certain three-class association schemes and ternary linear codes with dual distance three. In particular, we obtain a family of partial geometric designs from the three-class association schemes introduced by Kageyama, Saha, and Das in their article [“Reduction of the number of associate classes of hypercubic association schemes,” *Ann Inst Statist Math* **30** (1978)]. We also give a list of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.

Difference systems of sets (DSSs) are combinatorial structures arising in connection with code synchronization that were introduced by Levenshtein in 1971, and are a generalization of cyclic difference sets. In this paper, we consider a collection of *m*-subsets in a finite field of prime order to be a regular DSS for an integer *m*, and give a lower bound on the parameter ρ of the DSS using cyclotomic numbers. We show that when we choose -subsets from the multiplicative group of order *e*, the lower bound on ρ is independent of the choice of subsets. In addition, we present some computational results for DSSs with block sizes and , whose parameter ρ attains or comes close to the Levenshtein bound for .

Let *M* be an array (), where each of its cells is colored in one of two colors. We give a necessary and sufficient condition for the existence of a partition of *M* into *n* diagonals, each containing at least one cell of each color. As a consequence, it follows that if each color appears in at least cells, then such a partition exists. The proof uses results on completion of partial Latin squares.

Generalizing a result by Buratti et al.[M. Buratti, F. Rania, and F. Zuanni, Some constructions for cyclic perfect cycle systems, Discrete Math 299 (2005), 33–48], we present a construction for *i*-perfect *k*-cycle decompositions of the complete *m*-partite graph with parts of size *k*. These decompositions are sharply vertex-transitive under the additive group of with *R* a suitable ring of order *m*. The construction works whenever a suitable *i-perfect map* exists. We show that for determining the set of all triples for which such a map exists, it is crucial to calculate the chromatic numbers of some auxiliary graphs. We completely determine this set except for one special case where is the product of two distinct primes, is even, and . This result allows us to obtain a plethora of new *i*-perfect *k*-cycle decompositions of the complete graph of order (mod 2*k*) with *k* odd. In particular, if *k* is a prime, such a decomposition exists for any possible *i* provided that .

In this paper, by employing linear algebra methods we obtain the following main results:

- (i)Let and be two disjoint subsets of such that Suppose that is a family of subsets of such that for every pair and for every
*i*. ThenFurthermore, we extend this theorem to*k*-wise*L*-intersecting and obtain the corresponding result on two cross*L*-intersecting families. These results show that Snevily's conjectures proposed by Snevily (2003) are true under some restricted conditions. This result also gets an improvement of a theorem of Liu and Hwang (2013). - (ii)Let
*p*be a prime and let and be two subsets of such that or and Suppose that is a family of subsets of [*n*] such that (1) for every pair (2) for every*i*. ThenThis result improves the existing upper bound substantially.