Super-simple balanced incomplete block designs with block size 4 and index 5
A group divisible design (GDD) is a triple \((X, G, B)\), which satisfies the following conditions: {\parindent=6mm \begin{itemize}\item[1.] \(G\) is a partition of a set \(X\) (of points) into subsets called groups. \item[2.] \(B\) is a set of subsets of \(X\) (called blocks) such that a group and a block contain at most one common point. \item[3.] Every pair of points from distinct groups occurs in exactly \(\lambda\) blocks. \end{itemize}} The group type of a GDD is the multi-set \(\{|G|: G \in G\}\). A GDD with block sizes from a set of positive integers \(K\) is called a \((K, \lambda)\)-GDD, when \(K = \{k\}\), we use \(k\) for \(K\) ad if \(\lambda=1\), it is a \(k\)-GDD. A \((k, \lambda)\)-GDD with group type 1 is called a pairwise balanced design (PBD). A \((k, \lambda)\)-GDD with group type \(1v\) is called balanced incomplete block design (BIBD) is denoted by \((v, k, \lambda)\)-BIBD or \((v, b, r, k, \lambda)\) with \(v\) elements, \(b\) blocks, each element repeating in \(r\) blocks, each block is of size \(k\) and each pair of elements occurring in \(\lambda\) blocks. The necessary conditions for the existence of a BIBD are \(b \geq v\), \(bk = rv\), \(\lambda (v-1) = r (k-1)\). A transversal design \(TD (k, \lambda; n)\) is a \((k, \lambda)\)-GDD of a group type \(nk\) and block size \(k\), when \(\lambda=1\), we write \(TD (k,n)\), which is equivalent to \(k-2\) mutually orthogonal Latin squares (MOLS) of order \(n\). A design is called simple if it contains no repeated blocks. A design is said to be super-simple if the intersection of any two blocks has at most two elements. When \(k=3\), a super simple design is just a simple design. When \(\lambda=1\) the designs are necessarily super-simple. In this paper the authors addressed the existence problem of the super-simple BIBDs with \(k \geq 4\) and \(\lambda > 1\). Actually the term of super-simple designs was introduced by Gronau and Mullin [12]. The existence of super-simple designs is an interesting extremal problem by itself, but there are many useful applications in perfect harsh families, coverings, in the construction of new designs, and in the construction of super-imposed codes. Even in the statistical planning of experiments these super-simple designs provide samples with a maximum intersection as small as possible. The necessary conditions for the existence of a super-simple \((v, k, \lambda)\)-BIBD are {\parindent=6mm \begin{itemize}\item[1.] \(v \geq (k-2) \lambda +2\), \item[2.] \(\lambda (v-1) \equiv 0\pmod {k-1}\), \item[3.] \(\lambda v(v-1) \equiv 0\pmod {k(k-1)}\). \end{itemize}} And for an arbitrary \(v\) and \(\lambda\) these necessary conditions are asymptotically sufficient. For the existence of a super-simple \((v, 4, \lambda)\)-BIBD the necessary conditions are known to be sufficient for \(\lambda= 2, 3, 4, 6\). Gronau and Mullin [12] solved the case for \(\lambda= 2\) and corrected the proof of it. And the proof for \(\lambda= 3\), was given by Khodkar [16] and Chen [7]. The \(\lambda= 6\) case was solved by Chen, Cao and Wei [8]. A recent survey on super-simple \((v, 4, \lambda)\)-BIBDs with \(v \leq 32\) and all admissible \(\lambda\)s can be found in Bluskov and Heinrich [5]. All these results can be consolidated as one Theorem 1.2 as A super-simple \((v, 4, \lambda)\)-BIBD exists for \(\lambda= 2, 3, 4, 6\) if and only if the following conditions are satisfied {\parindent=6mm \begin{itemize}\item[1.] \(\lambda = 2, v \equiv 1\pmod 3\) and \(v \geq 7\), \item[2.] \(\lambda = 3, v \equiv 0, 1\pmod 3\) and \(v \geq 8\), \item[3.] \(\lambda = 4, v \equiv 1\pmod 3\) and \(v \geq 10\), \item[4.] \(\lambda = 6, v \geq 14\). \end{itemize}} What remained is the case \(\lambda = 5, v = 13\), which is going to be established by the authors in this paper by using the recursive constructions and certain other results. The authors investigated the existence of super-simple \((v, 4, 5)\)-BIBDs. Clearly when \(k=4\), and \(\lambda = 5\) the necessary conditions \(v \equiv 1, 4\pmod{12}\) and \(v \geq 13\). Now consolidating the existence results the authors mentioned two important theorems namely Theorem 3.10. A super-simple \((v, 4, 5)\)-BIBD exists for any \(v \equiv 1\pmod {12}\) and \(v \geq 13\). Theorem 4.11. A super-simple \((v, 4, 5)\)-BIBD exists for any \(v \equiv 4\pmod {12}\) and \(v \geq 16\). Finally now by combining these two theorems the authors established the main theorem of the paper namely Theorem 4.12. A super-simple \((v, 4, 5)\)-BIBD exists if and only if \(v \equiv 1, 4\pmod {12}\) and \(v \geq 13\).
scientific article; zbMATH DE number 5644070
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