Abstract A $$2-(v, k, \lambda )$$ 2 - ( v , k , λ ) design is additive if, up to isomorphism, the point set is a subset of an abelian group G and every block is zero-sum. This definition was introduced in Caggegi et al. (J Algebr Comb 45:271-294, 2017) and was the starting point of an interesting new theory. Although many additive designs have been constructed and known designs have been shown to be additive, these structures seem quite hard to construct in general, particularly when we look for additive Steiner 2-designs. One might generalize additive Steiner 2-designs in a natural way to graph decompositions as follows: given a simple graph $$\Gamma $$ Γ , an additive $$(K_v,\Gamma )$$ ( K v , Γ ) -design is a decomposition of the graph $$K_v$$ K v into subgraphs ( blocks ) $$B_1,\dots ,B_t$$ B 1 , ⋯ , B t all isomorphic to $$\Gamma $$ Γ , such that the vertex set $$V(K_v)$$ V ( K v ) is a subset of an abelian group G , and the sets $$V(B_1), \dots , V(B_t)$$ V ( B 1 ) , ⋯ , V ( B t ) are zero-sum in G . In this work we begin the study of additive $$(K_v,\Gamma )$$ ( K v , Γ ) -designs: we develop different tools instrumental in constructing these structures, and apply them to obtain some infinite classes of designs and many sporadic examples. We will consider decompositions into various graphs $$\Gamma $$ Γ , for instance cycles, paths, and k -matchings. Similar ideas will also allow us to present here a sporadic additive 2-(124, 4, 1) design.

Abstract A 2- $$(v,k,\lambda )$$ ( v , k , λ ) design is additive (or strongly additive) if it is possible to embed it in a suitable abelian group G in such a way that its block set is contained in (or coincides with) the set of all zero-sum k-subsets of its point set. Explicit results on the additivity or strong additivity of symmetric designs and subspace 2-designs are presented. In particular, the strong additivity of PG $$_d(n,q)$$ d ( n , q ) , which was known to be additive only for $$q=2$$ q = 2 or $$d=n-1$$ d = n - 1 , is always established.

Abstract Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v≡39 (mod 72), or $$v \equiv 4^e48 + 3$$ v≡4e48+3 (mod $$4^e96$$ 4e96 ) and $$e \ge 0$$ e≥0 , there exists a KTS onvpoints having at least $$v-3$$ v-3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism groupGacting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms ofG, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

Seven infinite classes of relative difference families with variable block sizes are presented explicitly. In particular, a balanced (gv,g,K,1)-DF withg=Σk∈K[(k2-k)/2] is explicitly given for: (i)K={3,4,5} and everyvcoprime to 6; (ii)K={3,4,6}, {3,5,6} or {3,4,5,6} and everyvcoprime to 30. As far as the authors are aware, these difference families can be viewed as the first explicit constructions of infinite classes of optimal variable-weight optical orthogonal codes with more than two weights. It is observed, however, that there are infinitely many values ofvfor which an optimal (v,W,1,Q) -OOC exists, whatever the set of weightsWand the weight distribution sequenceQare.

In this paper, a tight upper bound on the maximum possible code size of n, 4, 2, 1)-OOCs and some direct and recursive constructions of optimal (n, 4, 2, 1)-OOCs attaining the upper bound are given. As consequences, the following new infinite series of optimal (gn,4,2,1)-OOCs are obtained: i) g isin {1,7,11,19,23,31,35,59,71,79,131,179,191,239,251,271,311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971} or g is a primeh25i49ip1p2hellipprwhere h isin {0,1}, i and j are arbitrary nonnegative integers, and each piis a prime equiv 1 ( mod 8); ii) g = 2g' where g' isin {1,7,11,19,23,31,47,71,127,151,167,191,263,271,311,359,367,383,431,439,463,479,503,631,647,719,727,743,823,839,863,887,911,919,967,983,991} and n = p1p2hellipprwhere each piis a prime equiv 1 ( mod 4); iii) g isin {4,20} and n is any positive integer prime to 30; iv) g = 8 and n= p1p2hellipprwhere each piis a primary equiv 1 ( mod 4) greater than 5.