*i*. Then

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.

We construct all regular parallelisms of with automorphisms of order 3. Their number is 8. The two cyclic parallelisms found by Prince are among them. The other six ones are the first examples of noncyclic regular parallelisms and the first examples of regular parallelisms that do not belong to the infinite families of Penttila and Williams.

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

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.

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 .

An E–W matrix *M* is a ( − 1, 1)-matrix of order , where *t* is a positive integer, satisfying that the absolute value of its determinant attains Ehlich–Wojtas' bound. *M* is said to be of skew type (or simply skew) if is skew-symmetric where *I* is the identity matrix. In this paper, we draw a parallel between skew E–W matrices and skew Hadamard matrices concerning a question about the maximal determinant. As a consequence, a problem posted on Cameron's website [7] has been partially solved. Finally, codes constructed from skew E–W matrices are presented. A necessary and sufficient condition for these codes to be self-dual is given, and examples are provided for lengths up to 52.

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.

Quantum jump codes are quantum error-correcting codes which correct errors caused by quantum jumps. A *t*-spontaneous emission error design (*t*-SEED) was introduced by Beth et al. in 2003 [T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, A new class of designs which protect against quantum jumps, Des Codes Cryptogr 29 (2003), 51–70.] to construct quantum jump codes. The number of designs (dimension) in a *t*-SEED corresponds to the number of orthogonal basis states in a quantum jump code. A nondegenerate *t*-SEED is optimal if it has the largest possible dimension. In this paper, we investigate the bounds on the dimensions of 2-SEEDs systematically. The exact dimensions of optimal 2- SEEDs are almost determined, with five possible exceptions in doubt. General upper bounds on dimensions of 2- SEEDs are demonstrated, the corresponding leave graphs are described, and several exceptional cases are studied in details. Meanwhile, we employ 2-homogenous groups to obtain new lower bounds on the dimensions of 2- SEEDs for prime power orders *v* and general block sizes *k*.

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 .

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 .

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7-modular and 11-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a *p*-modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime *p*, we conditionally solve this conjecture and therefore the *p*-modular version of the Hadamard conjecture for all but finitely many cases when , and prove a weaker result for . Finally, we look at constraints on the existence of *m*-modular Hadamard matrices when the size of the matrix is small compared to *m*.

An elementary construction yields a new class of circulant (so-called “Butson-type”) generalized weighing matrices, which have order and weight *n*^{2}, all of whose entries are *n*th roots of unity, for all positive integers , where . The idea is extended to a wider class of constructions giving various group-developed generalized weighing matrices.

In this article, we show that if is a nontrivial nonsymmetric design admitting a flag-transitive point-primitive automorphism group *G*, then *G* must be an affine or almost simple group. Moreover, if the socle of *G* is sporadic, then is the unique 2 − (176, 8, 2) design with , the Higman–Sims simple group.