we denote by $ls[n](t,k,v)$ a large set of $t$-$(v,k,lambda)$ designs of size $n$, which is a partition of all  $k$-subsets ofa $v$-set into $n$ disjoint  $t$-$(v,k,lambda)$ designs and$n={v-t choose k-t}/lambda$.   we use the notation$n(t,v,k,lambda)$ as the maximum possible number of mutuallydisjoint cyclic $t$-$(v,k,lambda)$designs. in this paper we givesome new bounds for $n(2,29,4,3)$ and ...

In 1977, Ganter and Teirlinck proved that any 2t× 2t matrix with 2t nonzero elements can be partitioned into four submatrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any mt× nt matrix with kt nonzero elements can be partitioned into mn submatrices of order t of which at most k contain nonzero elements. In 1995, Brualdi et al. showed ...

we investigate the existence of some‎ ‎large sets of size nine‎. ‎the large set $ls[9](2,5,29)$ is constructed and‎ ‎existence of the family $ls[9](2,5,27l+j)$ for $lgeq 1, ‎2leq‎ ‎j

Kramer-Mesner matrices have been used as a powerful tool to construct t-designs. In this paper we construct Kramer-Mesner matrices for xed values of k and t in which the entries are polynomials in n the number of vertices of the underlying graph. From this we obtain an elementary proof that with a few exceptions S 2] n is a maximal subgroup of S (n 2) or A (n 2). We also show that there are onl...

A t-(v, k, λ) design is quasi-symmetric if there are only two block intersection sizes. We adapt the Kramer-Mesner construction method for designs with prescribed automorphism groups to the quasi-symmetric case. Using the adapted method, we find many new quasi-symmetric 2-(28, 12, 11) and 2-(36, 16, 12) designs, establish the existence of quasi-symmetric 2-(56, 16, 18) designs, and find three n...

Using the Kramer-Mesner method, 4-(26, 6, λ) designs with PSL(2, 25) as a group of automorphisms and with λ in the set {30, 51, 60, 81, 90, 111} are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called “quasi-designs”. Actions of groups PSL(2, 25), PGL(2, 25) and twisted PGL(2, 25) are being compared. It is shown that there exist 4-(2...

In this paper, we exploit some intersection matrices to empower a backtracking approach based on Kramer–Mesner matrices. As an application, we consider the interesting family of simple t-ðtþ 8; tþ 2; 4Þ designs, 1 t 4, and provide a complete classification for t 1⁄4 1; 4, as well as a classification of all non-rigid designs for t 1⁄4 2; 3. We also enumerate all rigid designs for t 1⁄4 2. The co...