Size-minimal combinatorial designs of staircase type
Barbora Batí ková, Tomáš J. Kepka, Petr C. Němec
TL;DR
The paper investigates size-minimal staircase-type combinatorial designs generated from partitions $n= r_1 s_1 + \dots + r_t s_t$ with $r_1 > \dots > r_t$. It develops a divisor-based framework using sets $\mathcal A(n)$ and $\mathcal B(n)$ to bound and optimize the key quantity $w= r_1 + s_1 + \dots + s_t$, introducing the central function $\varphi(n)$ that captures the minimum possible design size. A structural dichotomy is proved: if $n$ lies in the interval $I_k = \{k^2-k+1, \dots, k^2\}$ then $\varphi(n)=2k$, and if $n$ lies in $J_k = \{k^2+1, \dots, k^2+k\}$ then $\varphi(n)=2k+1$, with explicit constructions achieving these minima. The results yield a complete, elementary description of the minimal staircase design size in terms of $n$ and have implications for non-adaptive group testing by providing size-efficient designs with two symbol replications.
Abstract
Given a positive integer $n$ and a partitioning $n=r_1s_1+\dots+ r_ts_t$, $t,r_i,s_i$ positive integers, such that $r_1>\dots>r_t$ (for $t\ge 2$), we can write $n$ symbols $1,\dots,n$ in the form of a staircase matrix having $r_1$ rows where first $r_1-r_2$ rows have $x_1$ columns, next $r_2-r_3$ rows have $t_1+t_2$ columns, etc., and finally last $r_t$ rows have $t_1+\dots+t_k$ columns. Then we can construct a~design having $r_1+s_1+\dots+s_t$ sets by taking all $r_1$ rows and $s_1+\dots+s_t$ columns of this staircase matrix. Such designs have exactly two replications of each symbol and various cardinalities for the sets constituting the design. The minimum size of combinatorial designs of staircase type is found.