Some results on the Ryser design conjecture

TL;DR

This paper investigates properties of Ryser designs, providing new bounds and conditions under which these designs are of Type-1, and advances understanding of the Ryser-Woodall conjecture through theoretical results.

Contribution

The paper establishes new inequalities for block sizes, improves bounds on the order of Ryser designs, and confirms the Type-1 property under specific conditions, advancing the theoretical understanding of Ryser designs.

Findings

For every block, the size bounds are improved to $r_2 \\leq |A| \\leq r_1$.

Absence of small or large blocks implies $D \\leq -1$ or $D \\geq 0$ respectively.

Ryser designs with $2^n + 1$ points are proven to be of Type-1.

Abstract

A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points such that every two blocks intersect each other in $\lambda$ points (and $\lambda < v$ is a fixed number) and there are at least two block sizes. A design $\mathcal{D}$ is called a symmetric design, if every point of $\mathcal{D}$ has the same replication number (or equivalently, all the blocks have the same size) and every two blocks intersect each other in $\lambda$ points. The only known construction of a Ryser design is via block complementation of a symmetric design. Such a Ryser design is called a Ryser design of Type-1. This is the ground for the Ryser-Woodall conjecture: "every Ryser design is of Type-1". This long standing conjecture has been shown to be valid in many situations. Let $\mathcal{D}$ denote a Ryser design of order $v$, index $\lambda$ and replication numbers $r_1,r_2$. Let $e_i$ denote the number of points of $\mathcal{D}$ with replication number $r_i$ (with $i = 1, 2$). Call $A$ small (respectively large) if $|A| < 2\lambda$ (respectively $|A| > 2\lambda$) and average if $|A|=2\lambda$. Let $D$ denote the integer $e_1 - r_2$ and let $\rho> 1$ denote the rational number $\dfrac{r_1-1}{r_2-1}$. Main results of the present article are the following. For every block $A$, $r_1 \geq |A| \geq r_2$ (this improves an earlier known inequality $|A| \geq r_2$). If there is no small block (respectively no large block) in $\mathcal{D}$, then $D\leq -1$ (respectively $D\geq 0$). With an extra assumption $e_2 > e_1$ an earlier known upper bound on $v$ is improved from a cubic to a quadratic in $\lambda$. It is also proved that if $v \leq \lambda^2+ \lambda + 1$ and if $\rho$ equals $\lambda$ or $\lambda - 1$, then $\mathcal{D}$ is of Type-1. Finally a Ryser design with $ 2^n + 1$ points is shown to be of Type-1.

Equations

$$a<\lambda$$

$$e_1{r}_1(r_1-1)+e_2{r}_2(r_2-1)=\lambda v(v-1)$$

$$(\rho -1)e_1=\lambda (\rho +1)-r_2$$

$$e_1=\lambda +\frac{\lambda +D}{\rho }$$

$$(\rho -1)e_2=\rho r_1-\lambda (\rho +1)$$

$$e_2=\lambda +[\lambda -(D+1)]\rho$$

$$r_1=2\lambda +\left(\frac{a}{c}\right)(e_2-\lambda )$$

$$r_2=2\lambda -\left(\frac{a}{d}\right)(e_1-\lambda )$$

$$(r_1-1){\tau }_1(A)+(r_2-1){\tau }_2(A)=\lambda (v-1)$$

$${\tau }_1(A)=\lambda -td$$

$${\tau }_2(A)=\lambda +tc$$

$$|A|=2\lambda +ta$$

$$e_1{r}_1+e_2{r}_2=\lambda (v-1)+r_1{r}_2$$

$$x=\frac{e_2-\lambda }{c}=\frac{\lambda -(D+1)}{d}$$

$$y=\frac{e_1-\lambda }{d}=\frac{\lambda +D}{c}$$

$$x=\frac{\lambda -(D+1)}{d}$$

$$y=\frac{\lambda +D}{c}$$

$$xc+yd=v-2\lambda ,$$

$$xd+yc=2\lambda -1$$

$$x+y=g$$

$$x-y=\frac{v-(4\lambda -1)}{a}$$

$$D+1$$

$$D$$

$$e_1(gc+1)c+e_2(gd+1)d=\lambda [g(c+d)+1](c+d)$$

$$e_1(2^{m}c+1)c+e_2(2^{m}d+1)d=\lambda [2^n+1]2^k$$

Full text

For every block $A$, $r_{1}\geq|A|\geq r_{2}$ (this improves an earlier known inequality $|A|\geq r_{2}$). If there is no small block (respectively no large block) in $\mathcal{D}$, then $D\leq-1$ (respectively $D\geq 0$). With an extra assumption $e_{2}>e_{1}$ an earlier known upper bound on $v$ is improved from a cubic to a quadratic in $\lambda$. It is also proved that if $v\leq\lambda^{2}+\lambda+1$ and if $\rho$ equals $\lambda$ or $\lambda-1$, then $\mathcal{D}$ is of Type-1. Finally a Ryser design with $2^{n}+1$ points is shown to be of Type-1.

1. Introduction

A design is a pair $(X,L)$, where $X$ is a finite set of points and $L\subseteq P(X)$, where $P(X)$ is the power set of $X$. The elements of $X$ are called its points and the members of $L$ are called the blocks. Most of the definitions, formulas and proofs of standard results used here can be found in [3].

Definition 1.1**.**

A design $\mathcal{D}=(X,L)$ is said to be a symmetric $(v,k,\lambda)$ design if

$|X|=|L|=v$, 2. 2.

$|B_{1}\cap B_{2}|=\lambda\geq 1$ for all blocks $B_{1}$ and $B_{2}$ of $\mathcal{D},~{}~{}B_{1}\neq B_{2}$, 3. 3.

$|B|=k>\lambda$ for all blocks $B$ of $\mathcal{D}$.

Definition 1.2**.**

A design $\mathcal{D}=(X,L)$ is said to be a Ryser design of order $v$ and index $\lambda$ if it satisfies the following:

$|X|=|L|=v$, 2. 2.

$|B_{1}\cap B_{2}|=\lambda$ for all blocks $B_{1}$ and $B_{2}$ of $\mathcal{D},B_{1}\neq B_{2}$, 3. 3.

$|B|>\lambda$ for all blocks $B$ of $\mathcal{D}$, 4. 4.

there exist blocks $B_{1}$ and $B_{2}$ of $\mathcal{D}$ with $|B_{1}|\neq|B_{2}|$.

Here condition 4 distinguishes a Ryser design from a symmetric design, and condition 3 disallows repeated blocks and also any block being contained in another block.

Woodall [11] introduced a new type of combinatorial object which is a combinatorial dual of a Ryser design. All known examples of Ryser designs can be described by the following construction given by Ryser which is also known as the Ryser-Woodall complementation.

Let $\mathcal{D}=(X,\mathcal{A})$ be a symmetric $(v,k,k-\lambda)$ design with $k\neq 2\lambda$. Let $A$ be a fixed block of $\mathcal{D}$. Form the collection $\,\mathcal{B}=\{A\}\bigcup\{A\triangle B:B\in\mathcal{A},B\neq A\}$, where $A\triangle B$ denotes the usual symmetric difference of $A$ and $B$. Then $\overline{\mathcal{D}}=(X,\mathcal{B})$ is a Ryser design of order $v$ and index $\lambda$ obtained from $\mathcal{D}$ by block complementation with respect to the block $A$. We denote $\overline{\mathcal{D}}$ by $\mathcal{D}*A$. Then $A$ is also a block of $\mathcal{D}*A$ and the original design $\mathcal{D}$ can be obtained by complementing $\mathcal{D}*A$ with respect to the block $A$. If $\mathcal{D}$ is a symmetric $(v,k,\lambda^{{}^{\prime}})$ design, then the design obtained by complementing $\mathcal{D}$ with respect to some block is a Ryser design of order $v$ with index $\lambda=k-\lambda^{{}^{\prime}}$. A Ryser design obtained in this way is said to be of Type-1.

Define a Ryser design to be of Type-2 if it is not of Type-1. We now state

The Ryser Design Conjecture [3]: Every Ryser design is of Type-1.

In a significant paper Singhi and Shrikhande [9] proved the conjecture when the index $\lambda$ is a prime. In [8] Seress showed the truthfulness of the conjecture for $\lambda=2p$, where $p$ is a prime. In [2] Ionin and Shrikhande developed a new approach to the Ryser design conjecture that led to new results for certain parameter values. They also gave an alternate proof of the celebrated non-uniform Fisher Inequality. Ionin and Shrikhande went on to explore the validity of the Ryser design conjecture from a different perspective. Their results prove the conjecture for certain values of $v$ rather than for $\lambda$. Both Ryser and Woodall independently proved the following result:

Theorem 1.3** ([3, Theorem 14.1.2] Ryser Woodall Theorem).**

If $\mathcal{D}$ is a Ryser design of order $v$, then there exist integers $r_{1}$ and $r_{2}$, $r_{1}\neq r_{2}$ such that $r_{1}+r_{2}=v+1$ and any point occurs either in $r_{1}$ blocks or in $r_{2}$ blocks.

Let $\mathcal{D}$ be a Ryser design of order $v$ and index $\lambda$. It is known that a Ryser design has two invariants $r_{1}>r_{2}$ with $r_{1}+r_{2}=v+1$ such that every point has replication number either $r_{1}$ or $r_{2}$. Let $g$ denote the gcd between $r_{1}-1$ and $r_{2}-1$ and let $c$ and $d$ respectively denote the integers $\dfrac{r_{1}-1}{g}$ and $\dfrac{r_{2}-1}{g}$. Let $\rho=\dfrac{r_{1}-1}{r_{2}-1}=\dfrac{c}{d}$. Then $r_{1}+r_{2}=v+1$ implies $g\text{ divides }(v-1)\text{ and }v-1=(c+d)g$. Let $a=c-d$ then any two of $c,d$ and $a$ are coprime to each other. The point-set is partitioned into subsets $E_{1}$ and $E_{2}$, where $E_{i}$ is the set of points with replication number $r_{i}$ and let $e_{i}=|E_{i}|$ for $i=1,2$. Then $e_{1},e_{2}>0$ and $e_{1}+e_{2}=v$. For a block $A$, let $\tau_{i}(A)$ denote $|E_{i}\cap A|$, the number of points of block $A$ with replication number $r_{i}$ for $i=1,2$. Then $|A|=\tau_{1}(A)+\tau_{2}(A)$. We say a block $A$ is large, average or small if $|A|$ is greater than $2\lambda$, equal to $2\lambda$ or less than $2\lambda$ respectively. A block which is not average is called a non average block.

The Ryser-Woodall complementation of a Ryser design $\mathcal{D}$ of index $\lambda$ with respect to some block $\,A\in\mathcal{D}\,$ is either a symmetric design or a Ryser design of index $(|A|-\lambda)$. If $\mathcal{D}*A$ is the new Ryser design of index $(|A|-\lambda)$ obtained by Ryser-Woodall complementation of a Ryser design $\mathcal{D}$ with respect to the block $A$, we denote the new parameters of $\mathcal{\mathcal{D}*A}$ by $\lambda(\mathcal{D}*A),e_{1}(\mathcal{D}*A)$ etc.

Let $\mathcal{D}_{r}(X)$ denote the set of all incidence structures $\mathcal{D}=(X,\mathcal{B})$ where $\mathcal{B}$ is a set of subsets of $X$ and $\mathcal{D}$ is a Ryser design with replication numbers $r_{1}\text{ and }r_{2}=v+1-r_{1}$; or a symmetric design with block size $r_{1}\text{ or }r_{2}$.

Proposition 1.4** ([3, Proposition 14.1.7]).**

Let $\mathcal{D}\in\mathcal{D}_{r}(X)$ and let $A,B$ be blocks of $\mathcal{D}$. Then $\mathcal{D}*A\in\mathcal{D}_{r}(X)$ and the following conditions hold:

1. (i)

$(\mathcal{D}*A)*A=\mathcal{D}$; 2. (ii)

$A\triangle B$* is a block of $\mathcal{D}*A$ and $(\mathcal{D}*A)*(A\triangle B)=\mathcal{D}*(B)$;* 3. (iii)

$r_{1}(\mathcal{D}*A)=r_{1}(\mathcal{D})$; 4. (iv)

$\lambda(\mathcal{D}*A)=|A|-\lambda(\mathcal{D})$; 5. (v)

$E_{1}(\mathcal{D}*A)=E_{1}(\mathcal{D})\triangle A$; 6. (vi)

$e_{1}(\mathcal{D}*A)=e_{1}(\mathcal{D})-\tau_{1}(A)(\mathcal{D})+\tau_{2}(A)(\mathcal{D})$; 7. (vii)

$\mathcal{D}*A$* is a symmetric design if and only if $A=E_{1}(\mathcal{D})\text{ or }A=E_{2}(\mathcal{D})$.*

Remark 1.5**.**

Since $|A\triangle B|=|A|+|B|-2|A\cap B|$, observe that if a design is of Type-1 then it has all average blocks except for one, and hence a Type-2 Ryser design must have at least two non average blocks.

Theorem 1.6** ([3, Theorem 14.1.17]).**

For any Ryser design with index $\lambda>1$ and replication numbers $r_{1}\text{ and }r_{2}$, $\quad\displaystyle{\frac{\lambda}{\lambda-1}\leq\rho\leq\lambda}\text{ and }\rho\notin(\lambda-1,\lambda)$, where $\rho=\dfrac{r_{1}-1}{r_{2}-1}$.

Theorem 1.7** ([7, Theorem 1.7]).**

A Ryser design is of Type-1 if and only if $\,e_{1}e_{2}=\lambda(v-1)$.

Theorem 1.8** ([3, Theorem 14.4.8]).**

All Ryser designs of index less than 9 are of Type-1.

Theorem 1.9** ([3, Theorem 14.1.20]).**

If $r_{1},r_{2}$ are the replication numbers of a Ryser design of index $\lambda>2$, then $r_{1}-r_{2}\leq(\lambda-1)g,\text{ where }g=\gcd(r_{1}-1,r_{2}-1)$.

Theorem 1.10** ([3, Corollary 14.1.16]).**

If $\mathcal{D}$ is a Ryser design with $g=1$ , then $\mathcal{D}$ is of Type-1.

Theorem 1.11** ([3, Theorem 14.1.19]).**

The number of points of a Ryser design of index $\lambda>1$ does not exceed $\lambda^{3}+2$ and therefore for any fixed $\lambda>1$, there are only finitely many Ryser designs of index $\lambda$.

For any symmetric $(v,k,\lambda)$ design $\mathcal{D}$, the parameter $~{}n=k-\lambda$ is called the order of $\mathcal{D}$. It is well known that $4n-1\leq v\leq n^{2}+n+1~{}\text{ with }~{}v=4n-1$ if and only if $\mathcal{D}~{}\text{ is a Hadamard }(4n-1,2n-1,n-1)$ design or its complement, and $v=n^{2}+n+1\text{ if and only if }\mathcal{D}$ is a projective plane of order n or its complement. Ionin and Shrikhande [2] made the following conjecture.

Conjecture 1.12**.**

For any Ryser design on v points $\mbox{ }4\lambda-1\leq v\leq\lambda^{2}+\lambda+1$.

In this article, we show that for any block $A$ of a Ryser design, $r_{1}\geq|A|\geq r_{2}$. Let $D=e_{1}-r_{2}$. We establish a relation between the block size and the parameter $D$. We prove that if there is no small block (respectively no large block) in a Ryser design, then $D\leq-1\;(\text{respectively }\;D\geq 0)$. In support of Conjecture 1.12   we prove that $e_{2}>e_{1}$ implies $2\lambda^{2}+3\lambda-1>v\geq 4\lambda-1$. We also prove that Ryser designs with number of points at most $\lambda^{2}+\lambda+1$ and $\rho=\lambda$ or $\rho=\lambda-1$ are Type-1. Finally we prove that Ryser design with $v=2^{n}+1$ points is of Type-1.

2. A bound on the block sizes of a Ryser design

By Theorem 1.9   we have

[TABLE]

We use the following equations which can be found in [9] and[2].

[TABLE]

From equations (5) and (3), respectively:

[TABLE]

For a block $A$ with $|A|=\tau_{1}(A)+\tau_{2}(A)$, a simple two way counting gives

[TABLE]

Dividing through by $g$, the common g.c.d. of $r_{1}-1,r_{2}-1$ and $v-1$ yields: $c\tau_{1}(A)+d\tau_{2}(A)=\lambda(c+d)$ and hence $c(\tau_{1}(A)-\lambda)+d(\tau_{2}(A)-\lambda)=0.$ Using the coprimality of $c$ and $d$ then shows that $c$ divides $\tau_{2}(A)-\lambda$ while $d$ divides $\tau_{1}(A)-\lambda$ and writing the ratios to be $t$ and $s$ respectively, it is clear that $s=-t$ and hence $\tau_{1}(A)-\lambda=-td$ and $\tau_{2}(A)-\lambda=tc$ for some integer $t$. That is

[TABLE]

Hence we get the following lemma.

Lemma 2.1**.**

Let $A$ be a block of a Ryser design. Then the size of $A$ has the form $|A|=2\lambda+ta$, where $t$ is an integer. The block $A$ is large, average or small depending on whether $t>0,t=0$ or $t<0$ respectively. Hence $\tau_{1}(A)=\tau_{2}(A)=\lambda$ if $A$ is an average block, $\tau_{1}(A)>\lambda>\tau_{2}(A)$ if $A$ is a small block and $\tau_{2}(A)>\lambda>\tau_{1}(A)$ if $A$ is a large block.

Theorem 2.2**.**

Every block $A$ of a Ryser design has block size bounded by $r_{1}\geq|A|\geq r_{2}$.

Proof.

Since $\tau_{1}(A)\leq e_{1}$, equation (10) gives $-t\leq\dfrac{e_{1}-\lambda}{d}$, and hence we get $2\lambda+ta\geq 2\lambda-(e_{1}-\lambda)\dfrac{a}{d}$. By equations (12) and (8), we get $|A|\geq r_{2}$. Similarly, since $\tau_{2}(A)\leq e_{2}$, equation (11) gives $t\leq\dfrac{e_{2}-\lambda}{c}$, and hence $2\lambda+ta\leq 2\lambda+(e_{2}-\lambda)\dfrac{a}{c}$. Using equations (12) and (7), we get $|A|\leq r_{1}$. Therefore, we have $r_{1}\geq|A|\geq r_{2}$. ∎

3. Some results on the Ryser design conjecture

For a Ryser design with blocks $|A_{i}|=k_{i}\text{ for }i=1,2,\ldots,v$ the column sum of the incidence matrix is equal to the row sum of the incidence matrix which implies $\sum k_{i}=e_{1}r_{1}+e_{2}r_{2}$.

Hence from equation (3) and (5), we get

[TABLE]

In equation (7), let $x=\dfrac{e_{2}-\lambda}{c}$ and in equation (8), let $y=\dfrac{e_{1}-\lambda}{d}$. Then, $r_{1}=2\lambda+xa$ and $r_{2}=2\lambda-ya$. Since $c$ and $a$ are co-prime, it follows at once that $c$ divides $e_{2}-\lambda$ and hence $x$ is an integer. By equation (6) we get

[TABLE]

The assertion that $y$ is an integer follows similarly. By equation (4) we get

[TABLE]

From equations (14) and (15) we have:

[TABLE]

This gives us the following equations:

[TABLE]

[TABLE]

On adding equations (16) and (17) we get $x(c+d)+y(c+d)=v-1$. Since $~{}v-1=(c+d)g~{}$ we have

[TABLE]

Subtracting equation (17) from equation (16) obtains $x(c-d)-y(c-d)=v-(4\lambda-1)$. Hence we have,

[TABLE]

Lemma 3.1**.**

Let $\mathcal{D}$ be a Ryser design of order $v$ and index $\lambda$. Then the following conditions are equivalent.

1. (i)

$v\geq 4\lambda-1$** 2. (ii)

$x\geq y$** 3. (iii)

$e_{2}>e_{1}$.

Theorem 3.2**.**

The number of points of a Ryser design of index $\lambda>1$ does not exceed $2\lambda^{2}+3\lambda-1$ and therefore for any fixed $\lambda>1$, there are only finitely many Ryser designs of index $\lambda$. Further $e_{2}>e_{1}$ implies $2\lambda^{2}+3\lambda-1>v\geq 4\lambda-1$.

Proof.

Using equation (19), we have $(x-y)a=v-(4\lambda-1)$ and hence $v\geq 4\lambda-1$ if and only if $x\geq y$. Also, $~{}~{}v=4\lambda-1+(x-y)a~{}~{}$ and from equation (1) we know that $a<\lambda$. Hence, $~{}v<4\lambda-1+(x-y)\lambda$. From equation (17), $0\leq x-y<xd+yc=2\lambda-1$. Therefore, $~{}~{}v<4\lambda-1+(x-y)\lambda<4\lambda-1+(2\lambda-1)\lambda$, that is, $v<2\lambda^{2}+3\lambda-1$. Lemma 3.1   completes our proof. ∎

Proposition 3.3**.**

If there is no small block (respectively no large block) in a Ryser design, then $D\leq-1\;(\text{respectively }\;D\geq 0)$.

Proof.

From equation (13), $r_{1}(e_{1}-r_{2})+e_{2}r_{2}=\lambda(v-1)$. This gives $\,\,r_{1}D+e_{2}r_{2}=\lambda v-\lambda\text{ which implies }r_{1}D+\sum\limits_{A}(\tau_{2}(A))=\sum\limits_{A}(\lambda)-\lambda\text{ and hence }r_{1}D=\sum\limits_{A}(\lambda-\tau_{2}(A))-\lambda$. If there is no small block in the Ryser design then all the blocks are large or average. Using Lemma 2.1,$~{}~{}\tau_{2}(A)\geq\lambda$ for a large or an average block $A$ and hence if we do not have a small block then $D\leq-1$. Proof in the other case is similar. ∎

Theorem 3.4**.**

A Ryser design is of Type-1 if and only if $D=0\text{ or }D=-1$. We have the following cases for different values of $D$.

1. (i)

$D=0$* if and only if $yc=\lambda\text{ and }xd=\lambda-1$.* 2. (ii)

$D=-1$* if and only if $xd=\lambda\text{ and }yc=\lambda-1$.* 3. (iii)

$D>0\text{ if and only if }yc>\lambda>\lambda-1>xd$. 4. (iv)

$D<-1\text{ if and only if }xd>\lambda>\lambda-1>yc$.

Proof.

Equations (4) and (6) imply $e_{1}e_{2}=\lambda(v-1)-D(D+1)$. By Theorem 1.7 a Ryser design is of Type-1 if and only if $\,\,D(D+1)=0$. That is a Ryser design is of Type-1 if and only if $\,\,D=0\,\,\text{or}\,\,D=-1$.

Equations (14) and (15) imply

[TABLE]

Then all the four statements follow from the two equations above. ∎

By Theorem 1.6,   we have $~{}~{}\rho\leq\lambda\text{ and }\rho\notin(\lambda-1,\lambda)$. Therefore we consider the cases with $\rho=\lambda$ and $\rho=\lambda-1$.

Lemma 3.5**.**

Let $\mathcal{D}$ be a Type-2 Ryser design with $\rho=\lambda$ or $\rho=\lambda-1$. Then $D<-1$.

Proof.

We prove the result for $\rho=\lambda-1$. The case $\rho=\lambda$ is similar.

Since $\rho=\dfrac{c}{d},\text{ with }\gcd(c,d)=1$ we have $\,c=\lambda-1,~{}~{}~{}d=1$ and hence, $a=c-d=\lambda-2$. By remark 1.5,$~{}~{}\mathcal{D}$ has at least two non average blocks. Complement $\mathcal{D}$ with respect to any non average block of size $k$ to obtain a new Ryser design $\overline{\mathcal{D}}$. Let $\overline{\rho}=\dfrac{\overline{r_{1}}-1}{\overline{r_{2}}-1}$, where $\overline{r_{1}}\text{ and }\overline{r_{2}}$ are the replication numbers and let $\overline{\lambda}$ be the index of the new design. By Proposition 1.4 (iii)   and Theorem 1.6,   we get $\overline{\lambda}=k-\lambda\geq\overline{\rho}=\rho=\lambda-1,$ for the new design which implies $k\geq 2\lambda-1$. That is, a non average blocks of $\mathcal{D}$ is either large or small of size $2\lambda-1$.

Since a small block has size $k=2\lambda-ta$ for some positive integer $t$ we have, $~{}2\lambda-ta=2\lambda-1\text{ which implies }t=1$ and $a=1$. That is, $~{}a=\lambda-2=1\text{ and hence }\lambda=3$. By Theorem 1.8,   $\mathcal{D}$ is of Type-1, a contradiction.

Therefore, a non average block of $\mathcal{D}$ is large and by Proposition 3.3, we have $D\leq-1$. Using the fact that $\mathcal{D}$ is of Type-2 Theorem 3.4   now implies that $D<-1$ completing the proof. ∎

Theorem 3.6**.**

Let $\mathcal{D}$ be a Ryser design with at most $\lambda^{2}+\lambda+1$ points with $\rho=\lambda$ or $\rho=\lambda-1$. Then, $\mathcal{D}$ is of Type-1.

Proof.

Suppose not. Then $\mathcal{D}$ is of Type-2. We prove that this leads to a contradiction.

Let $\rho=\lambda$. Then $\lambda=\dfrac{c}{d},\text{ with }\gcd(c,d)=1$ implies $\,c=\lambda,~{}~{}~{}d=1$ and hence, $a=c-d=\lambda-1$. By lemma (3.5) we have $D<-1$. Equations (16) and (17) imply $x\lambda+y=v-2\lambda$ and $x+y\lambda=2\lambda-1$. Adding the two we get $(x+y)(\lambda+1)=v-1$. Then, by equation (18) we have $g=\dfrac{v-1}{\lambda+1}$. Now, $g=x+y=x+\lambda y-y(\lambda-1)$. Then, by equation (17) we get $\dfrac{v-1}{\lambda+1}=2\lambda-1-y(\lambda-1)$. This gives us $\lambda^{2}+\lambda+1-v=(\lambda^{2}-1)(y-1)$, and hence $y\geq 1$. Since $D<-1$, Theorem 3.4 (iv)   implies $x>\lambda>\lambda-1>y(\lambda-1)~{}$, with $y\geq 1$ which is impossible. The case $\rho=\lambda-1$ is similar. ∎

Theorem 3.7**.**

A Ryser design with $v=2^{n}+1$ points is of Type-1.

Proof.

Note right at the beginning that the case $r_{2}=1$ leads to $r_{1}=v$ forcing repeated blocks that are disallowed by the stipulations on a Ryser design. Equation (2) and division by $g$, then obtains:

[TABLE]

By Theorem 1.10   we are done if $g=1$ and hence we must assume that $g\neq 1$. Since $g$ must divide $v-1=2^{n}$ and $g\neq v-1$, we have: $g=2^{m}$ where $1\leq m\leq n-1$ is an integer. Let $k=m-n$. Then equation (22) reduces to:

[TABLE]

Here, $k$ is a positive integer showing that the right side is an even number. Since $c+d=2^{k}$, it follows that $c,d$ have the same parity and since their g.c.d. is $1$, $c$ and $d$ must be both odd. On the other hand, $e_{1}+e_{2}=2^{n}+1$. So exactly one of $e_{1}$ and $e_{2}$ is odd say $e_{1}$ is odd and $e_{2}$ is even. On the left side of equation (23), the first summand is odd while the second is even forcing the right side to be odd, which is impossible. ∎