Some new designs with prescribed automorphism groups

TL;DR

This paper proves the existence of several previously unknown simple combinatorial designs with specific parameters, expanding the catalog of known designs and analyzing the conditions for their existence.

Contribution

It establishes the existence of new simple designs with various parameters and investigates the set of lambda values for which such designs exist.

Findings

Existence of simple 2-(55,10,4) design

Existence of simple 3-(20,5,4) and 3-(21,7,30) designs

Existence of simple 4-(15,5,2), 4-(16,8,45) designs, and 5-(16,7,10), 5-(17,8,40) designs

Abstract

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding $t$, $v$, and $k$, we study the set of all $\lambda$ for which simple $t$-$(v,k,\lambda)$ designs exist.

Tables

Table 1. Table 1. Base blocks for 3 3 3 - ( 20 , 5 , λ ) 20 5 𝜆 (20,5,\lambda) designs, λ ∈ { 4 , 10 , 14 } 𝜆 4 10 14 \lambda\in\{4,10,14\} .

$\{1,2,3,5,13\}$	$\{1,2,3,8,10\}$	$\{1,2,4,6,12\}$	$\{1,2,4,8,17\}$	$\{1,2,4,10,20\}$
$\{1,2,5,7,15\}$	$\{1,2,5,9,16\}$	$\{1,2,6,8,20\}$	$\{1,2,6,9,11\}$	$\{1,2,7,11,14\}$
$\{1,3,6,9,15\}$	$\{1,4,9,13,20\}$
$\{1,2,3,5,12\}$	$\{1,2,3,5,16\}$	$\{1,2,3,6,16\}$	$\{1,2,3,6,17\}$	$\{1,2,3,7,12\}$
$\{1,2,3,7,16\}$	$\{1,2,4,5,9\}$	$\{1,2,4,9,11\}$	$\{1,2,4,9,15\}$	$\{1,2,4,9,20\}$
$\{1,2,4,11,16\}$	$\{1,2,4,11,18\}$	$\{1,2,4,18,20\}$	$\{1,2,5,8,9\}$	$\{1,2,5,8,12\}$
$\{1,2,5,8,15\}$	$\{1,2,5,8,16\}$	$\{1,2,6,14,15\}$	$\{1,2,7,12,13\}$	$\{1,2,7,12,14\}$
$\{1,2,8,9,20\}$	$\{1,2,8,10,12\}$	$\{1,2,8,12,20\}$	$\{1,2,9,10,20\}$	$\{1,2,9,11,13\}$
$\{1,2,9,13,20\}$	$\{1,3,5,9,12\}$	$\{1,3,5,11,14\}$	$\{1,3,6,8,14\}$	$\{1,3,6,8,20\}$
$\{1,3,6,13,16\}$	$\{1,3,6,14,16\}$	$\{1,3,6,16,20\}$	$\{1,3,6,17,20\}$	$\{1,3,7,16,20\}$
$\{1,3,8,11,14\}$	$\{1,4,7,11,16\}$	$\{1,4,10,14,20\}$	$\{1,5,10,14,20\}$

Table 2. Table 2. Base blocks for a 3 3 3 - ( 21 , 7 , 30 ) 21 7 30 (21,7,30) design with G 1 subscript 𝐺 1 G_{1} and a 3 3 3 - ( 21 , 7 , 75 ) 21 7 75 (21,7,75) design with G 2 subscript 𝐺 2 G_{2} as automorphism group.

$\{1,2,3,4,5,6,7\}$	$\{1,2,3,7,8,13,17\}$	$\{1,2,3,7,9,11,14\}$
$\{1,2,3,7,9,12,21\}$	$\{1,2,7,8,10,16,21\}$	$\{1,2,7,8,11,14,20\}$
$\{1,2,7,10,16,19,20\}$	$\{1,7,8,12,13,18,19\}$	$\{1,7,8,15,18,19,21\}$
$\{7,8,9,10,12,15,19\}$
$\{1,2,3,4,7,8,20\}$	$\{1,2,3,4,7,14,15\}$	$\{1,2,3,7,8,10,12\}$
$\{1,2,3,9,13,15,20\}$	$\{1,2,7,8,9,10,16\}$	$\{1,2,7,8,9,13,19\}$
$\{1,2,7,8,9,15,18\}$	$\{1,2,7,8,11,14,17\}$	$\{1,2,7,9,12,17,20\}$
$\{1,2,7,10,11,16,17\}$	$\{1,2,7,10,11,19,20\}$	$\{1,2,7,10,16,19,20\}$
$\{1,7,8,9,10,12,15\}$	$\{1,7,8,9,17,19,21\}$	$\{7,8,9,10,12,13,21\}$
$\{7,8,9,10,17,18,20\}$	$\{7,8,9,13,17,19,21\}$

Table 3. Table 3. Base blocks for a 4 4 4 - ( 15 , 5 , 2 ) 15 5 2 (15,5,2) design.

$\{1,2,3,4,5\}$	$\{1,2,3,7,11\}$	$\{1,2,4,7,11\}$	$\{1,2,4,7,14\}$
$\{1,2,4,8,10\}$	$\{1,2,4,8,11\}$	$\{1,2,4,9,12\}$	$\{1,2,4,9,15\}$
$\{1,2,4,10,13\}$	$\{1,2,4,12,14\}$	$\{1,2,4,13,15\}$	$\{1,2,7,8,9\}$
$\{1,2,7,8,14\}$	$\{1,2,7,12,15\}$	$\{1,2,10,11,12\}$	$\{1,2,10,11,15\}$
$\{1,2,13,14,15\}$	$\{1,4,7,8,12\}$	$\{1,4,7,9,11\}$	$\{1,4,8,9,14\}$
$\{1,4,8,12,13\}$	$\{1,4,8,13,14\}$	$\{1,4,9,10,11\}$	$\{1,4,11,12,15\}$
$\{1,4,11,14,15\}$	$\{1,7,8,10,11\}$	$\{1,7,8,10,13\}$	$\{1,7,8,11,15\}$
$\{1,7,8,14,15\}$	$\{1,7,10,11,14\}$	$\{1,7,10,13,15\}$	$\{1,7,11,12,14\}$
$\{1,7,12,13,14\}$	$\{1,10,11,13,15\}$	$\{7,8,9,10,13\}$	$\{7,8,11,12,13\}$

Table 4. Table 4. Base blocks for a 4 4 4 - ( 16 , 8 , 45 ) 16 8 45 (16,8,45) design.

$\{1,2,3,4,5,6,8,11\}$	$\{1,2,3,4,5,6,14,16\}$	$\{1,2,3,4,5,8,9,14\}$
$\{1,2,3,4,5,8,12,13\}$	$\{1,2,3,4,5,10,12,14\}$	$\{1,2,3,4,5,11,15,16\}$
$\{1,2,3,4,6,8,10,16\}$	$\{1,2,3,4,6,13,15,16\}$	$\{1,2,4,5,6,7,9,11\}$
$\{1,2,4,5,6,7,13,15\}$	$\{1,2,4,5,6,9,11,15\}$	$\{1,2,4,5,6,9,11,16\}$
$\{1,2,4,5,6,10,13,16\}$	$\{1,2,4,5,7,8,9,16\}$	$\{1,2,4,5,8,12,15,16\}$
$\{1,2,4,6,8,9,10,15\}$

Table 5. Table 5. Base blocks for a 5 5 5 - ( 16 , 7 , 10 ) 16 7 10 (16,7,10) design.

$\{1,2,3,4,5,6,13\}$	$\{1,2,3,4,5,6,14\}$	$\{1,2,3,5,6,7,11\}$	$\{1,2,3,5,6,8,9\}$
$\{1,2,3,5,6,9,10\}$	$\{1,2,3,5,6,9,12\}$	$\{1,2,3,5,6,10,15\}$	$\{1,2,3,5,6,14,16\}$
$\{1,2,3,5,8,11,12\}$	$\{1,2,5,6,7,8,16\}$	$\{1,2,5,6,7,9,14\}$	$\{1,2,5,6,7,12,13\}$
$\{1,2,5,6,7,14,15\}$

Table 6. Table 6. Base blocks for a 5 5 5 - ( 17 , 8 , 40 ) 17 8 40 (17,8,40) design.

$\{1,2,3,4,5,6,7,10\}$	$\{1,2,3,4,5,6,8,10\}$	$\{1,2,3,4,5,6,8,13\}$
$\{1,2,3,4,5,6,9,12\}$	$\{1,2,3,4,5,6,10,14\}$	$\{1,2,3,4,5,7,8,12\}$
$\{1,2,3,4,5,7,9,14\}$	$\{1,2,3,4,5,7,10,14\}$	$\{1,2,3,4,5,7,11,13\}$
$\{1,2,3,4,5,7,14,15\}$	$\{1,2,3,4,5,9,10,12\}$	$\{1,2,3,4,6,7,8,9\}$
$\{1,2,3,4,6,7,9,16\}$	$\{1,2,3,4,6,7,12,16\}$	$\{1,2,3,4,6,7,14,16\}$
$\{1,2,3,4,6,8,14,16\}$	$\{1,2,3,4,6,9,11,15\}$	$\{1,2,3,4,7,8,9,12\}$

Equations

$$(1,2,\dots ,11)(12,\dots ,22)(23,\dots ,33)(34,\dots ,44)(45,\dots ,55)$$

$$\begin{array}[]{l}(2,27)(3,37)(4,29)(5,47)(6,53)(7,30)(8,38)(9,32)(11,43)(13,52)\\ (15,48)(17,28)(18,42)(19,49)(20,51)(21,44)(22,31)(23,46)(24,50)\\ (25,54)(26,33)(34,45)(36,39)(41,55).\end{array}$$

$$\begin{array}[]{l}(3,23)(4,20)(5,10)(6,45)(7,34)(8,33)(11,12)(13,26)(14,15)(16,53)\\ (17,43)(18,28)(19,52)(21,49)(22,47)(24,46)(25,32)(27,51)(29,42)\\ (31,41)(36,39)(38,48)(40,54)(44,50).\end{array}$$

$$(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11).$$

$$(2,3,4,5,6)(7,8,9,10,11)(12,13,14,15,16)(17,18,19,20,21)$$

$$(1,5,2)(3,4,6)(9,18,16)(10,13,12)(11,14,19)(15,17,20).$$

$$(1,4)(2,6)(3,5)(10,12)(11,14)(15,20)(16,18)$$

$$(4,5,6)(7,8,9)(10,11,12)(13,14,15),$$

$$(1,4)(2,5)(3,6)(8,13)(9,10)(11,15).$$

$$(2,3)(4,5,6,7)(8,9,10,11)(12,13,14,15),$$

$$(1,5)(2,13)(3,11)(6,15)(7,8)(9,14)(10,12).$$

$$(2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16),$$

$$(1,5)(2,12)(3,15)(4,8)(6,14)(7,16)(9,10)(11,13).$$

$$(2,4,10,11,14,6,16,12,17,15,9,8,5,13,3,7).$$

Full text

The author is partially supported by the Croatian Science Foundation under project 1637.

1. Introduction

A $t$-$(v,k,\lambda)$ design $\mathcal{D}$ is a $v$-element set $\mathcal{P}$ of points together with a collection $\mathcal{B}$ of $k$-element subsets called blocks, such that every $t$-element subset of points is contained in exactly $\lambda$ blocks. The design is simple if $\mathcal{B}$ is a set, i.e. contains no repeated blocks. We refer to the Handbook of Combinatorial Designs [4] for definitions and results about designs. In this paper we are concerned with the existence problem for simple designs with small parameters.

An automorphism of $\mathcal{D}$ is a permutation of points leaving $\mathcal{B}$ invariant. The set of all automorphisms forms a group under composition, the full automorphism group $\mathop{\mathrm{Aut}}(\mathcal{D})$. By prescribing suitable subgroups $G\leq\mathop{\mathrm{Aut}}(\mathcal{D})$, we are able to construct simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$. These designs are designated as unknown in [12, Table 1.35] and [7, Table 4.46]. Since the Handbook was published, new existence results about designs with small parameters have appeared in [1], [2], [15], [16], and [17]. To the best of our knowledge, existence of the constructed designs has previously been open.

The layout of our paper is as follows. In the next section we describe the construction method, the used computational tools, and some other preliminary matters. Sections 3 to 9 are dedicated to designs with particular $t$, $v$, and $k$. We present our new constructions and try to determine the set of all $\lambda$ such that simple $t$-$(v,k,\lambda)$ designs exist. This is accomplished for $4$-$(15,5,\lambda)$ designs in Section 6, and in the other sections up to three open cases remain. The prescribed groups $G$ and some computational details are laid out in the proofs of the theorems. Designs are presented by listing base blocks; $\mathcal{B}$ is the union of the corresponding $G$-orbits. The groups and the base blocks for the new designs are also available on our web page:

https://web.math.pmf.unizg.hr/~krcko/results/newdesigns.html

2. Preliminaries

Let $G$ be a group of permutations of $\mathcal{P}=\{1,\ldots,v\}$, and let $\mathcal{T}_{1},\ldots,\mathcal{T}_{m}$ and $\mathcal{K}_{1},\ldots,\mathcal{K}_{n}$ be the orbits of $t$-element subsets and $k$-element subsets of $\mathcal{P}$, respectively. Let $a_{ij}$ be the number of subsets $K\in\mathcal{K}_{j}$ containing a given $T\in\mathcal{T}_{i}$. This number does not depend on the choice of $T$. The matrix $A=[a_{ij}]$ is the Kramer-Mesner matrix. It is well known that simple $t$-$(v,k,\lambda)$ designs with $G$ as an automorphism group exist if and only if the system of linear equations $A\cdot x=\lambda j$ has [math]-$1$ solutions $x\in\{0,1\}^{n}$. Here, $j=(1,\ldots,1)^{\tau}$ is the all-one vector of length $m$. This method of construction was pioneered by E. S. Kramer and D. M. Mesner [8] and has since been used to find many new designs with prescribed automorphism groups (see, e.g., [2], [3], [9], [10], [11], and [18]).

We use GAP [6] to compute the orbits $\mathcal{T}_{i}$, $\mathcal{K}_{j}$ and the matrix $A$. For most of our results this is a small and easy computation. Exceptions are Theorems 3.1 and 3.2, where we use an algorithm described in [10] to produce short orbits, and a program written in C for long orbits of the group $G_{3}\cong M_{11}$. The second step of the computation is finding solutions of the Kramer-Mesner system $A\cdot x=\lambda j$. Solving systems of linear equations over $\{0,1\}$ is a known NP complete problem. Our prescribed automorphism groups $G$ lead to systems of small to moderate size, that can be solved fairly quickly. We use the backtracking solver developed in [9] for $2$-designs and A. Wasserman’s solver [18] based on the LLL algorithm for designs with $t\geq 3$. Running times varied from a few seconds to several days of CPU time. Finally, to decide whether the constructed designs are isomorphic and to compute their full automorphism groups, we use nauty/Traces by B. D. McKay and A. Piperno [13].

A $t$-$(v,k,\lambda)$ design $\mathcal{D}$ is also a $s$-$(v,k,\lambda_{s})$ design, for $\lambda_{s}=\lambda{v-s\choose t-s}/{k-s\choose t-s}$, $0\leq s\leq t$. The number of blocks of $\mathcal{D}$ is $b=|\mathcal{B}|=\lambda_{0}$. The parameters $t$-$(v,k,\lambda)$ are called admissible if all the $\lambda_{s}$ are integers, and realizable if simple designs with these parameters exists. Given $t$, $v$, and $k$, there is a least integer $\lambda_{\min}$ such that $t$-$(v,k,\lambda_{\min})$ are admissible. Any $\lambda$ for which $t$-$(v,k,\lambda)$ are admissible is of the form $\lambda=m\lambda_{\min}$, for $m\in\mathbb{N}$. The largest $\lambda$ for which a simple $t$-$(v,k,\lambda)$ design exists is $\lambda_{\max}={v-t\choose k-t}$. The corresponding complete $t$-$(v,k,\lambda_{\max})$ designs contains all $k$-element subsets as blocks: $\mathcal{B}={\mathcal{P}\choose k}$.

For the parameters of our newly constructed designs, we try to determine all $\lambda$ between $\lambda_{\min}$ and $\lambda_{\max}$ such that $t$-$(v,k,\lambda)$ are realizable. The supplement of a $t$-$(v,k,\lambda)$ design $\mathcal{D}=(\mathcal{P},\mathcal{B})$ is the design $\overline{\mathcal{D}}=(\mathcal{P},{\mathcal{P}\choose k}\setminus\mathcal{B})$ with parameters $t$-$(v,k,\lambda_{\max}-\lambda)$. Therefore it suffices to consider existence of $t$-$(v,k,\lambda)$ designs for $\lambda\leq\lambda_{\max}/2$. We shall denote the largest integer $m$ such that $m\lambda_{\min}\leq\lambda_{\max}/2$ by $M$. The complement of $\mathcal{D}$, obtained by taking the complement of each block in $\mathcal{P}$, is also a $t$-design and therefore it suffices to consider parameters with $k\leq v/2$.

3. Designs with parameters $2$-$(55,10,\lambda)$

Let $t=2$, $v=55$, and $k=10$. Then $\lambda_{\min}=1$, $\lambda_{\max}={53\choose 8}=886322710$, and $M=443161355$. A $2$-$(55,10,1)$ design would have $b=33$ blocks and cannot exist by Fisher’s inequality [12, Theorem 1.9]. Designs $2$-$(55,10,2)$ are quasi-residuals of symmetric $2$-$(67,12,2)$ designs, which do not exist by the Bruck-Ryser-Chowla theorem [4, Theorem II.6.11]. By the Hall-Connor theorem [4, Theorem II.6.27], quasi-residual designs with $\lambda=2$ are actually residual, hence $2$-$(55,10,2)$ designs also do not exist. According to [12, Table 1.35], $2$-$(55,10,m)$ designs exist for $m=5$, and are unknown for $m\in\{3,4,6\}$.

Theorem 3.1**.**

Simple $2$-$(55,10,4)$ designs exist.

Proof.

Let $G_{1}\cong PSL(2,11)$ be the group of order $660$ generated by the permutations

[TABLE]

and

[TABLE]

The group $G_{1}$ has $6$ orbits on $2$-element subsets of $\mathcal{P}=\{1,\ldots,55\}$. It suffices to consider orbits of $10$-element subsets whose size does not exceed the number of blocks $b=132$. This can be accomplished efficiently by an algorithm described in [10]; there are $97$ such orbits. The $6\times 97$ Kramer-Mesner system has $5$ solutions for $\lambda=4$, giving rise to three non-isomorphic designs. Two of them have $G_{1}$ as their full automorphism group. They are generated by the base blocks $\{1,2,5,8,10,12,23,42,45,50\}$ and $\{1,2,3,9,18,27,28,33,38,43\}$, respectively. The third design has $\mathop{\mathrm{Aut}}(\mathcal{D})\cong G_{1}.\mathbb{Z}_{2}\cong PGL(2,11)$ and is generated by the two base blocks $\{1,2,4,5,16,18,24$, $30,51,55\}$ and $\{1,2,6,8,24,27,38,40,50,53\}$. ∎

Designs $2$-$(55,10,5)$ can be constructed from the group $G_{2}\cong\mathbb{Z}_{55}.\mathbb{Z}_{10}$. There are five non-isomorphic designs with $G_{2}$ as their full automorphism group.

Theorem 3.2**.**

Simple $2$-$(55,10,m)$ designs exist for $m\in\{4,5,8,\ldots,M\}$.

Proof.

For $m\in\{8,\ldots,58\}$, designs can be constructed from the group $G_{1}$. For example, $2$-$(55,10,8)$ designs are obtained as unions of any two of the designs from Theorem 3.1, since they are disjoint. By the Kramer-Mesner method we found designs for all $m$ in the specified range. Base blocks are available on the web page referred to in the Introduction.

For $m\geq 59$, we use the group $G_{3}\cong M_{11}$ of order $7920$ generated by the permutations (1) and

[TABLE]

Orbits of size less than $|G_{3}|$ were computed by the algorithm from [10]. There are $367$ orbits of size less than $3960$. Using these orbits and the Kramer-Mesner method, we found designs for $m\in\{59,\ldots,479\}$. Base blocks are available on our web page. There are $13753$ orbits of size $3960$, of which $1647$ form the blocks of $2$-$(55,10,120)$ designs. These $1647$ designs are mutually disjoint and disjoint from the previously constructed designs. Finally, we used a program written in C to find the $3686048$ long orbits of size $7920$. Among them, $555578$ orbits form disjoint $2$-$(55,10,240)$ designs, and $1256273$ pairs of long orbits can be combined into as many disjoint $2$-$(55,10,480)$ designs. It is clear that by taking unions of the so far constructed designs, simple $2$-$(55,10,m)$ design with $G_{3}$ as an automorphism group can be constructed for any $m\in\{59,\ldots,M\}$. ∎

Recently, D. Crnković and A. Švob [5] also found $2$-$(55,10,m)$ designs for $m\in\{4,10\}$. Thus, the only remaining open cases are $m\in\{3,6,7\}$. We tried to construct these designs using various prescribed automorphism groups, but did not find any examples.

4. Designs with parameters $3$-$(20,5,\lambda)$

For $t=3$, $v=20$, and $k=5$, we have $\lambda_{\min}=2$, $\lambda_{\max}={17\choose 2}=136$, and $M=34$. According to [7, Table 4.46], $3$-$(20,5,2m)$ designs exist for $m\in\{3,\ldots,6,8,\ldots,34\}$. We found designs for two of the three missing values of $m$.

Theorem 4.1**.**

Simple $3$-$(20,5,2m)$ designs exist for $m\in\{2,7\}$.

Proof.

Let $G$ be the dihedral group of order $38$, generated by the cycle $(1,2,\ldots,19)$ and the involution

[TABLE]

There are $39$ orbits of $3$-element subsets and $444$ orbits of $5$-element subsets of $\mathcal{P}=\{1,\ldots,20\}$. One solution of the Kramer-Mesner system for $\lambda=4$ is given by the first $12$ base blocks in Table 1. The next $39$ base blocks generate a $3$-$(20,5,10)$ design, disjoint from the $3$-$(20,5,4)$ design. The union of these two design is a simple $3$-$(20,5,14)$ design. ∎

Corollary 4.2**.**

Simple $3$-$(20,5,2m)$ designs exist for $m\in\{2,\ldots,34\}$.

In fact, designs can be constructed from the same group $G$ for all $m\in\{2,\ldots,34\}$. The only open case is now $m=1$. We did not find any $3$-$(20,5,2)$ designs by prescribing automorphism groups. We examined the subgroups of $PGL(2,19)$ operating on $20$ points and of $AGL(1,19)$ operating on $19$ points, of orders greater than $19$, systematically.

5. Designs with parameters $3$-$(21,7,\lambda)$

For $t=3$, $v=21$, and $k=7$, we have $\lambda_{\min}=15$, $\lambda_{\max}={18\choose 4}=3060$, and $M=102$. According to [7, Table 4.46], $3$-$(21,7,15m)$ designs exist for $46$ values of $m$. Existence is unknown for $56$ values of $m$, starting with $m\in\{1,2,5,7,\ldots\}$. We found designs for all but the first of these unknown values.

Theorem 5.1**.**

Simple $3$-$(21,7,15m)$ designs exist for $m\in\{2,\ldots,102\}$.

Proof.

Let $G_{1}\cong A_{6}$ be the group of order $360$ generated by the permutations

[TABLE]

and

[TABLE]

There are $12$ orbits of $3$-element subsets and $406$ orbits of $7$-element subsets of $\mathcal{P}=\{1,\ldots,21\}$. The Kramer-Mesner system has $56$ solutions for $\lambda=30$, giving rise to $28$ non-isomorphic designs. All of them have $G_{1}$ as their full automorphism group. Base blocks for one of the designs are the first $10$ sets in Table 2.

The group $G_{2}\cong S_{6}$ of order $720$ generated by the permutations (2) and

[TABLE]

can be used for $m\geq 3$. The Kramer-Mesner system is of size $11\times 253$. We checked that solutions exist for all $\lambda=15m$, $m\in\{3,\dots,102\}$. Base blocks for $m=5$ are the next $17$ sets in Table 2. Base blocks for the other cases are available on our web page. ∎

We did not find any $3$-$(21,7,15)$ designs, and the existence problem is still open.

6. Designs with parameters

$4$-$(15,5,\lambda)$

For $t=4$, $v=15$, and $k=5$, we have $\lambda_{\min}=1$, $\lambda_{\max}=11$, and $M=5$. It is known that $4$-$(15,5,m)$ designs do not exist for $m=1$ [14] and exist for $m\in\{3,4,5\}$ [7, Table 4.46]. We settle the remaining case $m=2$.

Theorem 6.1**.**

Simple $4$-$(15,5,2)$ designs exist.

Proof.

Let $G\cong\mathbb{Z}_{3}\times S_{3}$ be the group of order $18$ generated by the permutations

[TABLE]

[TABLE]

The Kramer-Mesner system is of size $84\times 178$ and has $12$ solutions for $\lambda=2$, giving rise to two non-isomorphic designs with $\mathop{\mathrm{Aut}}(\mathcal{D})=G$. Base blocks for one of them are given in Table 3. ∎

For $m\in\{3,5\}$ designs can be constructed from the same group $G$, and for $m=4$ from the group $G_{1}\cong A_{5}$ of order $60$.

7. Designs with parameters $4$-$(16,8,\lambda)$

For $t=4$, $v=16$, and $k=8$, we have $\lambda_{\min}=15$, $\lambda_{\max}={12\choose 4}=495$, and $M=16$. By [7, Table 4.46], $4$-$(16,8,15m)$ designs exist for $m\in\{4,\ldots,16\}$. We found designs for $m=3$.

Theorem 7.1**.**

Simple $4$-$(16,8,45)$ designs exist.

Proof.

Let $G\cong\mathbb{Z}_{15}.(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ be the group of order $120$ generated by the permutations

[TABLE]

[TABLE]

The Kramer-Mesner system is of size $25\times 132$ and has four solutions for $\lambda=45$. They correspond to four non-isomorphic designs with $\mathop{\mathrm{Aut}}(\mathcal{D})=G$. Base blocks for one of them are given in Table 4. ∎

The same group $G$ can be used to construct $4$-$(16,8,15m)$ designs for $m\in\{4,\ldots,16\}$. We tried many groups for $m\in\{1,2\}$, but did not find any designs.

8. Designs with parameters $5$-$(16,7,\lambda)$

For $t=5$, $v=16$, and $k=7$, we have $\lambda_{\min}=5$, $\lambda_{\max}={11\choose 2}=55$, and $M=5$. By [7, Table 4.46], $5$-$(16,7,5m)$ designs exist for $m\in\{3,4,5\}$. Here we settle the case $m=2$.

Theorem 8.1**.**

Simple $5$-$(16,7,10)$ designs exist.

Proof.

Let $G\cong(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}).A_{4}$ be the group of order $192$ generated by the permutations

[TABLE]

[TABLE]

The Kramer-Mesner system is of size $28\times 71$ and has two solutions for $\lambda=10$. The two designs are isomorphic and have $\mathop{\mathrm{Aut}}(\mathcal{D})=G$. Base blocks are listed in Table 5. ∎

The same group $G$ gives designs for $m=5$, and for $m\in\{3,4\}$ a subgroup of index $2$ can be used. We did not find any designs for $m=1$.

9. Designs with parameters $5$-$(17,8,\lambda)$

For $t=5$, $v=17$, and $k=8$, we have $\lambda_{\min}=20$, $\lambda_{\max}={12\choose 3}=220$, and $M=5$. By [7, Table 4.46], $5$-$(17,8,20m)$ designs exist for $m\in\{3,4,5\}$. Again, we can settle the $m=2$ case.

Theorem 9.1**.**

Simple $5$-$(17,8,40)$ designs exist.

Proof.

Let $G\cong\mathbb{Z}_{17}.\mathbb{Z}_{16}$ be the group of order $272$ generated by the cycle $(1,2,\ldots,17)$ and the permutation

[TABLE]

The Kramer-Mesner system is of size $25\times 95$. It has $61$ solutions for $\lambda=40$, giving rise to $61$ non-isomorphic designs with $\mathop{\mathrm{Aut}}(\mathcal{D})=G$. Base blocks for one of the designs are given in Table 6. ∎

The same group can be used for $m\in\{3,4,5\}$. The existence of $5$-$(17,8,20)$ designs remains open.