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We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed. Necessary existence conditions are proved and a table of feasible parameters of such configurations with at most 200 points is presented. Non-existence of some configurations with feasible parameters is proved., 23 pages, 1 figure. Revision: added Proposition 5.6 and made some minor corrections

We study 4-designs with three intersection numbers. By the Cameron-Delsarte theorem, the blocks form a symmetric three-class association scheme. This imposes strong restrictions on the parameters of such designs. We calculate the eigenvalues of the association scheme from the design parameters and determine all admissible parameters with at most 1000 points. An infinite family of admissible parameters is discovered. Designs with small admissible parameters exist and are related to the quadratic residue codes.

Quasi-symmetric (36, 16, 12) designs with intersection numbers $$x=6$$ , $$y=8$$ that cannot be embedded in symmetric (64, 28, 12) designs as residuals are constructed.

We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common features: combinatorial interpretations, Pascal-like recurrences, inverse relations with their signed versions, and interpretations as coefficients of change between polynomial bases., Comment: 10 pages

We construct a new partial geometry with parameters pg(5,5,2), not isomorphic to the partial geometry of van Lint and Schrijver., Comment: 5 pages

Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with $56$ points. One new $(56,16,18)$ and many new $(56,16,6)$ designs are discovered, and non-existence of $(56,12,9)$ and $(56,20,19)$ designs with certain automorphism groups is proved. The number of known symmetric $(78,22,6)$ designs is also significantly increased., Comment: 18 pages

We establish the existence of simple designs with parameters $2$-$(55, 10, 4)$, $3$-$(20, 5, 4)$, $3$-$(21, 7, 30)$, $4$-$(15, 5, 2)$, $4$-$(16, 8, 45)$, $5$-$(16, 7, 10)$, and $5$-$(17, 8, 40)$, which have previously been unknown. For the corresponding $t$, $v$, and $k$, we study the set of all $\lambda$ for which simple $t$-$(v, k, \lambda)$ designs exist.

We prove the existence of designs with parameters 4-(18, 9, 56), 4-(18, 9, 70), 4-(19, 9, 84), and 4-(19, 9, 105) by assuming suitable automorphism groups and performing a computer search using the method of Kramer and Mesner.

Equations for coefficients of tactical decomposition matrices for 2-designs are well-known and they have been used for constructions of many examples of 2-designs. In this paper, we generalize these equations and propose an explicit equation system for coefficients of tactical decomposition matrices for $$t\text{-}(v,k,\lambda _t)$$ designs, for any integer value of $$t.$$

Napoleon’s quasigroups are idempotent medial quasigroups satisfying the identity (ab·b)(b·ba) = b. In works by V. Volenec geometric terminology has been introduced in medial quasigroups, enabling proofs of many theorems of plane geometry to be carried out by formal calculations in a quasigroup. This class of quasigroups is particularly suited for proving Napoleon’s theorem and other similar theorems about equilateral triangles and centroids.

We propose a combination of two known computational methods for the construction of designs with prescribed groups of automorphisms: the Kramer–Mesner method and the method of tactical decompositions. This combined method is used to construct new unitals with parameters 2-(65, 5, 1). © 2011 Wiley Periodicals, Inc. J Combin Designs 19:290-303, 2011

A t-(v, k, lambda) design is quasi-symmetric if any two blocks intersect either in x or in y points, for non-negative integers x

We study tilings of groups with mutually disjoint difference sets. Some necessary existence conditions are proved and shown not to be sufficient. In the case of tilings with two difference sets we show the equivalence to skew Hadamard difference sets, and prove that they must be normalized if the group is abelian. Furthermore, we present some constructions of tilings based on cyclotomy and investigate tilings consisting of Singer difference sets.

It is known that the classical unital arising from the Hermitian curve in PG(2,9) does not have a 2-coloring without monochromatic lines. Here we show that for q≥4 the Hermitian curve in PG(2,q2) does possess 2-colorings without monochromatic lines. We present general constructions and also prove a lower bound on the size of blocking sets in the classical unital.

U ovom članku predstavljamo Java applet za interaktivni prikaz trodimenzionalnih objekata izrađenih u programskom sustavu Mathematica. Autor appleta je Martin Kraus s Tehničkog sveučilišta u Münchenu, a applet je uključen u on-line inačicu Mathematice, webMathematicu. Sam applet slobodno je dostupan i možete se njime koristiti da biste postavili interaktivne trodimenzionalne objekte na vlastite web stranice. U članku objašnjavamo kako se to radi, a za čitatelje koji se ne žele upuštati u izradu web stranica pokazujemo kako postaviti trodimenzionalne objekte na forumu math.e.

TSC(k, l) is defined as a finite linear space i.e. an incidence structure with v points such that any two of them are joined by a unique line, consisting of two symmetric configurations - (v_k) and (v_l), k and l being the "lengths" of lines. E.g. each (9_3) configuration , including the Pappus configuration, may be extended in an obvious manner to a TSC(2, 3). Here we describe a construction method based on projective planes which possibly yields an infinite class of TSC spaces, depending on the number of solutions of a certain open problem from number theory. We also present some other constructions of TSC spaces, including a full classification of TSC(3, 4).

In this article designs with parameters S(2, 4, 28) and nontrivial automorphism groups are classified. A total of 4466 designs were found. Together with some S(2, 4, 28)'s with trivial automorphism groups found by A.Betten, D.Betten and V.D.Tonchev this sums up to 4653 nonisomorphic S(2, 4, 28) designs.