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On the existence of (v, k, t) trades
TLDR
It is proved that, given integers v > k > t ~ 1, there does not exist a (v, k, t) trade of volume s, where 2t < s < 2t + 2t-l.
On the possible volume of three way trades
On the 3-way (v, k, 2) Steiner trades
RP2: a high-performance data center network architecture using projective planes
TLDR
RP2 is underlain by the use of multi-port servers and mini-switches in DCN development and the application of a recursive method for scaling up a data center and constitutes a high-performance and highly scalable DCN architecture.
Smallest defining sets of super-simple 2 - (v, 4,1) directed designs
A $2-(v,k,\lambda)$ directed design (or simply a $2-(v,k,\lambda)DD$) is super-simple if its underlying $2-(v,k,2\lambda)BIBD$ is super-simple, that is, any two blocks of the $BIBD$ intersect in at
ON THE POSSIBLE VOLUME OF -( v; k; t ) TRADES
A µ-way (v,k,t) trade of volume m consists of µ disjoint collections T1, T2,...Tµ, each of m blocks, such that for every t-subset of v-set V the number of blocks containing this t-subset is the same
On defining sets of directed designs
TLDR
Some lower bounds for the number of blocks in smallest defining sets in directed designs are introduced, the precise number ofblocks in smallest defines sets for some directed designs with small parameters are determined and an open problem is pointed out relating to thenumber of blocks needed to define a directed design as compared with the number needed to defined its underlying undirected design.
On directed trades
TLDR
This study shows that the volume of a (v, k, t)DT is at least 2Lt/ 2J and that directed trades with minimum volume and minimum foundation exist.
On the existence of $d$-homogeneous $3$-way Steiner trades
A $\mu$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_{\mu} \}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$, each of $m$ blocks of size $k$, such that
On the non-existence of some Steiner $t$-$(v,k)$ trades of certain volumes
Mahmoodian and Soltankhah $\cite{MMS}$ conjectured that there does not exist any $t$-$(v,k)$ trade of volume $s_{i}< s <s_{i+1}$, where $s_{i}=2^{t+1}-2^{t-i}, i=0,1,..., t-1$. Also they showed that
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