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\title{\bf\boldmath Existence of 2-$(v,k,1)$\ designs admitting\\
a block-transitive group of affine type}

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\author{Ding Shifeng, Liu Weijun\thanks{Supported by the
National Natural Science Foundation of China(Grant No.
11271208).}\\
\small Department of Mathematics\\[-0.8ex]
\small Central South University\\[-0.8ex]
\small Changsha 410083, China\\
\small\tt isgibt@yahoo.com.cn}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Jun 8, 2012}{Sep 19, 2012}{Sep 27, 2012}\\
\small Mathematics Subject Classifications: 05B05, 20B25}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
% succinct statement of background followed by a listing of the
% principal new results that are to be found in the paper. The abstract
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\begin{abstract}
In this paper we use Weil's estimate on character sums to prove
that large admissible prime powers $q$ admit 2-$(q,k,1)$ designs
having block-transitive automorphism groups in AGL$(1, q)$.


% keywords are optional
\bigskip\noindent \textbf{Keywords:} Design; Block-transitive; Weil's
theorem
\end{abstract}

\section{Introduction}
A 2-$(v,k,1)$ design $\mathcal{D}$ is a system
$(\mathcal{P},\mathcal{B})$, where $\mathcal{P}$ is a set of $v$
points and $\mathcal{B}$ is a collection of some $k$-subsets of
$\mathcal{P}$, called blocks, such that any two different points
from $\mathcal{P}$ lie on exactly one block $B\in{\mathcal B}$. A
flag is a pair $(\alpha,B)$ where $\alpha$ is a point and $B$ a
block containing $\alpha$.

An automorphism of $\mathcal{D}$ is a permutation of the point set
$\mathcal{P}$ which preserves the incidence relation. The set of
automorphisms of $\mathcal{D}$ is denoted by
$\textnormal{Aut}\mathcal{D}$.  Let
$G\leq\textnormal{Aut}\mathcal{D}$. If $G$ acts transitively on
the block set $\mathcal{B}$ of $\mathcal{D}$, then $G$ is said to
be block-transitive. Similarly, if $G$ acts transitively on the
flags of $\mathcal{D}$, then $G$ is said to be flag-transitive.


The classification of the 2-$(v, k, 1)$ designs with
flag-transitive automorphism groups has been completed
~\cite{bue}. Recently, the designs with block- transitive
automorphism groups are of great interest
(see~\cite{cam1,cam2,han,lih}). In \cite{liandliu}, H. L. Li and
W. J. Liu prove that if a soluble group $G$ acts on a design
$\mathcal D$ block-transitively, and suppose the point size of
$\mathcal D$ satisfies $v > (k^{3/4} + 1)^{\phi(k(k-1))}$, then
$v$ is a power of a prime $p$ and $G$ is flag-transitive or $G\leq
\textnormal{A}\Gamma\textnormal{L}(1, v)$.

In this paper, we investigate the existence of the pairs
$(\mathcal{D},G)$ such that $\mathcal{D}$ is a 2-$(v, k, 1)$
design, $G$ is a one-dimensional affine group acting on $\mathcal
D$ as an automorphism group block-transitively. We show that there
is a method to construct such a pair $(\mathcal{D},G)$ from some
suitable prime power $q$. The main result is the following
theorem.


\begin{theorem}
  \label{Thm:Mainresult}
\ \ For every integer $k\geq3$, there exists a positive integer
$N(k)$ such that if $q$ is a power of a prime $p_0>k-2$ satisfying
that $q>N(k)$ and  $q\equiv k(k-1)+1(\bmod\ 2k(k-1))$, then there
exists a 2-$(q,k,1)$ design $\mathcal{D}$, which has a regular
block-transitive automorphism group $G<\textnormal{AGL}(1,q)$.
\end{theorem}

In Theorem 1, the bound $N(k)\approx
2^{k(k-1)}k^4(k(k-1)/2)^{2k-2}$. This bound is by no means a good
one, therefore Theorem 1 only shows that such a bound exists. When
$k$ is small, we can find a much better bound. We will prove
$N(3)=1$ in section 4. For those prime powers $q \leq N(k)$ in the
theorem, if $q$ is not large, we can search the designs with the
aid of computers. For example, for $k=4$ it is shown that
$N(4)\approx10^4$ in \cite{lin}, but by using some simple programs
to search the required blocks \cite{ding2}, we find that the
designs exist for each prime power $q\leq5000$.

We explain Theorem 1 briefly. There are many examples of
2-$(v,k,1)$ designs with block-transitive (or flag-transitive)
automorphism groups of 1-dimensional affine type, such as
translation affine planes, generalized Netto systems, and Kantor's
inflation trick, etc. But a complete classification of these
designs  seems to be out of reach with present methods.  On the
other hand, R. M. Wilson's classic paper  \cite{wilson} states
that if $q$ is a prime power greater than $(k(k-1)/2)^{k(k-1)}$,
then there exists a $(q,k,\lambda)$-difference family in
$\textnormal{GF}(q)$ if and only if $\lambda(q-1)\equiv0(\bmod \
k(k-1))$, where $\lambda$ is a positive integer. We believe there
is an important and intriguing connection between Wilson's work
and the construction of designs with block-transitive (or
flag-transitive) automorphism groups of affine type. Similar to
Wilson's result, Theorem 1 shows that large admissible prime
powers $q$ admit Steiner 2-designs having block-transitive
automorphism groups in AGL$(1, q)$. We use Weil's estimate on
character sums to get the bound $N(k)$. When $k\geq5$, our bound
is  smaller than Wilson's bound $(k(k-1)/2)^{k(k-1)}$, which is
obtained by other techniques. But we have assumed an additional
condition $p_0>k-2$ in order to use Weil's Theorem in a simple
way.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Some preliminary results}

Throughout this paper, we assume $k\geq3$ is an integer, $p_0>k-2$
is a prime, and $q=p_0^n$ is a power of $p_0$  such that $q\equiv
k(k-1)+1\ (\bmod{\ 2k(k-1)})$. Since $k(k-1)+1$ and $2k(k-1)$ are
relatively prime, by Dirichlet's theorem on primes in arithmetic
progressions, there are infinitely many primes $p_0$ (hence
infinitely many $q$) satisfying the above requirements. For an
arbitrary set $X$, let $|X|$ denote its cardinality.

Let $\textnormal{GF}(q)$ be the finite field of $q$ elements and
let $\theta$ be a generating element of the multiplicative group
$\textnormal{GF}(q)^\times$. Let $M$ and $L$ be the subgroups of
$\textnormal{GF}(q)^\times$ defined by
\[M=\left<\theta^{k(k-1)/2}\right>, L=\left<\theta^{k(k-1)}\right>.\]
Then clearly, $[M:L]=2$.

Let $G\subseteq\textnormal{AGL}(1,q)$ be the group of all
permutations of $\textnormal{GF}(q)$ of the form $x\rightarrow
\alpha x+\sigma$, where $\alpha\in L$,
$\sigma\in\textnormal{GF}(q)$. Then clearly
$G=\textnormal{GF}(q)^+:L$.

Let $B=\{\beta_1,\beta_2,\cdots, \beta_k\}$ be a subset of
$\textnormal{GF}(q)$ consisting of $k$ pairwise distinct elements.
Define $\triangle{ B}=\{\beta_j-\beta_i|1\leq i<j\leq k\}$.
Clearly $|\triangle B|\leq k(k-1)/2$. For an element $g\in G$,
define $B^g=\{\beta_1^g,\beta_2^g,\cdots,\beta_k^g\}$. Let
$B^G=\{B^g|g\in G\}$.



\begin{lemma}{\textnormal{(\cite{ding})}} Let
$B=\{\beta_1,\beta_2,\cdots,\beta_k\}$ be a $k$-subset of
$\textnormal{GF}(q)$.  If $\triangle B$ is exactly a system of
representatives of the cosets of $M$ in
$\textnormal{GF}(q)^\times$, then
$\mathcal{D}=(\textnormal{GF}(q),B^G)$ is a 2-$(q,k,1)$ design,
and $G$ is regular on the blocks of $\mathcal {D}$.

\end{lemma}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weil's theorem on character sums}
In this section, we introduce the notion of multiplicative
characters of finite fields, then we quote the character sum
version of Weil's theorem. We mention that various applications of
Weil's theorem in finite geometry and combinatorics have been
surveyed in $\cite{szo}$.

Let $F$ be a finite field and let $\mathbb{C}$ be the set of
complex numbers. A multiplicative character of $F$ is a
homomorphism $\chi:F^\times\rightarrow\mathbb{C}^\times$.  The
{\em trivial character} $\chi_0$ is defined by $\chi_0(\alpha)=1$
for all $\alpha\in F^\times$. Let $\chi_1$ and $\chi_2$ be two
multiplicative characters of $F$. Then define $\chi_1\chi_2$ to be
the map
\[\chi_1\chi_2(\alpha)=\chi_1(\alpha)\chi_2(\alpha), \ \ \
\forall \alpha\in F^\times.\] This  definition makes the set of
multiplicative characters of $F$ into a group. This group is a
cyclic group of order $|F|-1$. A character $\chi$ is said to be of
order $m$ if $m$ is the least positive integer such that
$\chi^m=\chi_0$. It is often useful to extend the domain of
definition of a multiplicative character to all of $F$. If $\chi$
is not the trivial character, we do this by defining $\chi(0)=0$.
For the trivial character $\chi_0$, we define $\chi_0(0)=1$.\\


\begin{lemma}\ Let $G^*$ be a group of multiplicative characters of a
finite field $F$ and let $\alpha$ be a fixed element of $
F^\times$. Then we have

\[\sum_{\chi\in G^*}\chi(\alpha)=\left\{
\begin{array}{ll}|G^*|,&{\textnormal {if}}\ \chi(\alpha)=1\ {\textnormal{ for all}}\ \ \chi\in G^*;\\
0,&{\textnormal{if}}\ \chi(\alpha)\not=1 \ \ {\textnormal{for some
}}\chi\in G^*.\end{array}\right.
\]
\end{lemma}

\begin{proof} Clearly, if $\chi(\alpha)=1$ for all $\chi\in G^*$,
then we have $\sum_{\chi\in G^*}\chi(\alpha)=|G^*|$. Suppose there
is a character, say $\psi\in G^*$, such that $\psi(\alpha)\not=1$,
then we have

\[\psi(\alpha)\sum_{\chi\in G^*}\chi(\alpha)=\sum_{\chi\in G^*}\psi(\alpha)\chi(\alpha)
=\sum_{\chi\in G^*}\psi\chi(\alpha)=\sum_{\chi\in
G^*}\chi(\alpha).\] The last equality follows since $\psi\chi$
runs over all characters of $G^*$ as $\chi$ does. The result
follows immediately.
\end{proof}

\noindent{\bf Weil's Theorem}.\ (See \cite{fin},Theorem 5.41)\ \
{\em Let $\psi$ be a multiplicative character of a finite field
$F$ and suppose $\psi$ is of order $m>1$. Suppose that $f\in F[x]$
is a monic polynomial of positive degree, and that $f$ is not an
$m$th power of a polynomial. Let $d$ denote the number of distinct
roots of $f$ in its splitting field over $F$. Then for any element
$\alpha\in F$, we have
\[\left|\sum_{x\in\scriptsize{ F}}
\psi(\alpha f(x))\right|\leq (d-1)\sqrt{|F|}.\] }


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem 1}
We need some simple properties of the cyclotomic polynomials. The
$n$th cyclotomic polynomial $\Phi_n(x)$ is defined by

\[\Phi_n(x)=\prod_{(j,n)=1}(x-e^{\frac{2\pi i}{n}j}),
\ \  \textnormal{where}\ 1\leq j\leq n. \] It is well known that
$\Phi_n(x)\in\mathbb{Z}[x]$.\\

\begin{lemma}{\textnormal{\cite{fin}}}\ \
$x^{n}-1=\prod_{d|n}\Phi_d(x)$.
\end{lemma}

\begin{lemma}Let $m$ and $n$ be different positive integers,
and let $p$ be a prime with $p\nmid mn$. Then as polynomials in
$\textnormal{GF}(p)[x]$, $\Phi_m(x)$ and $\Phi_n(x)$ are
relatively prime.
\end{lemma}

\begin{proof} Consider the polynomial $x^{mn}-1$. This polynomial
has no multiple roots in any extension field of
$\textnormal{GF}(p)$ since $(p,mn)=1$. By Lemma 4,
$\Phi_n(x)\Phi_m(x)$ divides $x^{mn}-1$, thus they have no
nontrivial common factors.
\end{proof}

By Lemma 2,  if we can choose a set $B=\{0,1,\beta,$ $ \beta^2$, $
\beta^3,\cdots,\beta^{k-2}\}$, where $\beta\in\textnormal{GF}(q)$,
such that $\triangle B$ is a system of representatives of the
cosets of $M$ in $\textnormal{GF}(q)^\times$, then a  2-$(v,k,1)$
design on which $G=\textnormal{GF}(q)^+: L$ is regular
block-transitive can be constructed. Thus the idea is to show that
for any fixed integer $k\geq3$, if $q$ is large enough, such an
element $\beta\in\textnormal{GF}(q)$ always exists.

We list the elements of $\triangle B$ in the following table. We
use the symbol $\textnormal{C}j$ ($j\in\{1,2,\cdots,k-1\}$) to
denote the $j$th column of the table.

\begin{center}{\small {\bf Table}:\ The elements of $\triangle B$ }

\[\begin{array}{|llllll|}\hline
\textnormal{\small C1}&\textnormal{\small
C2}&\textnormal{C3}&\cdots&
\textnormal{\small C(k-2)}&\textnormal{\small C(k-1)}\\
\hline
1&\beta-1&\beta^2-1&\cdots&\beta^{k-3}-1&\beta^{k-2}-1\\
\beta &\beta(\beta-1)&\beta(\beta^2-1)&\cdots&\beta(\beta^{k-3}-1)&\\
\beta^2&\beta^2(\beta-1) &\beta^2(\beta^2-1)&\cdots&&\\
\vdots &\vdots &\vdots &&&\\
\beta^{k-4}&\beta^{k-4}(\beta-1)&\beta^{k-4}(\beta^2-1)&&&\\
\beta^{k-3}&\beta^{k-3}(\beta-1)&&&&\\
\beta^{k-2}&&&&&\\
\hline
\end{array}\]
\end{center}

\begin{lemma}\  Let
$B=\{0,1,\beta,\beta^2,\cdots,\beta^{k-2}\}$. Suppose that
$\beta\in\textnormal{GF}(q)$ satisfies the following conditions
\begin{equation}\left\{\begin{array}{l}\beta\in M\theta,\\
\beta-1\in M\theta^{k-1},\\
\beta^2-1\in M\theta^{2k-3},\\
\ \ \ \ \vdots\\
\beta^j-1\in M\theta^{jk-(1+2+3+\cdots+j)},\\
(j=1,2,\cdots,k-2).\end{array}\right.
\end{equation}Then $\triangle B$ is a system of representatives of the
cosets of $M$ in $\textnormal{GF}(q)^\times$.
\end{lemma}

\begin{proof} The cosets of $M$ in $\textnormal{GF}(q)^\times$ are
$M\theta^j$, where $j=0,1,\cdots,\frac{k(k-1)}{2}-1$.

If $\beta\in M\theta$, then the elements in the first column C1
run through $M$, $M\theta$, $\cdots$, $M\theta^{k-2}$. While from
$\beta-1\in M\theta^{k-1}$ we know the elements in the second
column C2 run through $M\theta^{k-1}$, $M\theta^k$, $\cdots$,
$M\theta^{2k-4}$. It is not hard to verify that  for $j=$
1,2,$\cdots$,$k-2$, the elements in C$j$ run through
$M\theta^{(j-1)k-(1+2+\cdots \underline{j-1})}$, $\cdots$,
$M\theta^{jk-(1+2+\cdots+j)-1}$. And finally, we have
$\beta^{k-2}-1\in
M\theta^{(k-2)k-(1+2+\cdots+\underline{k-2})}=M\theta^{\frac{k(k-1)}{2}-1}$.

We note that in (1) the arrangement of the the coset which
$\beta^j-1$ belongs to  does not contradict the arrangements of
the cosets for $\beta^i-1$ $(i < j)$. In fact, we only need to
specify the cosets which $\beta$, $\Phi_1(\beta)$,
$\Phi_2(\beta)$, $\cdots$, $\Phi_{k-2}(\beta)$ belong to one by
one in turn.  We have assumed $p_0>k-2$. By Lemma 5, $\Phi_j(x)$
is relatively prime to $x\Phi_1(x)\Phi_2(x)\cdots\Phi_{j-1}(x)$,
thus the arrangements for $\Phi_i(\beta)$ $(i<j)$ do not affect
the arrangement for $\Phi_j(\beta)$. \end{proof}


Now $\beta^{j}-1\in M\theta^{jk-(1+2+\cdots+j)}$ is equivalent to
$\theta^{-jk+(1+2+\cdots+j)}(\beta^j-1)$ $\in M$. For any element
$\alpha\in\textnormal{GF}(q)^\times$, we have
$\alpha^{k(k-1)/2}\in M$. Thus the conditions in (1) can be stated
in another way.

\begin{equation}\left\{\begin{array}{l}\beta\in M\theta,\\
\beta^{k(k-1)/2-k+1}(\beta-1)\in M,\\
\beta^{k(k-1)/2-2k+3}(\beta^2-1)\in M,\\
\ \ \ \ \vdots\\
\beta^{k(k-1)/2-jk+(1+2+\cdots+j)}(\beta^j-1)\in M,\\
(j=1,2,\cdots,k-2).\end{array}\right.
\end{equation}

We use (2) instead of (1), and introduce the following
polynomials,

\begin{equation}f_j(x)=x^{k(k-1)/2-jk+(1+2+\cdots+j)}(x^j-1), \ \
\forall j=1,2,\cdots, k-2.
\end{equation}
We consider these polynomials $f_j(x)$  in
$\textnormal{GF}(q)[x]$. Note that for a positive integer $ j\leq
k-2$, $jk-(1+2+\cdots+j)$ $=$ $(k-1)+(k-2)+\cdots+(k-j)$ is
greater that $j$ and less than $k(k-1)/2$, thus each polynomial
$f_j(x)$ is of a positive degree less than $k(k-1)/2$.

\begin{lemma}\   Let $m\geq2$ be an integer. Let
$g(x)=x^{j_0}f_1^{j_1}(x)\cdots f_{k-2}^{j_{k-2}}(x)$, where the
indexes $j_i$s are nonnegative integers,  not all zero, and each
$j_i<m$. Then as a polynomial in $\textnormal{GF}(q)[x]$, $g(x)$
is not an $m$th power of a polynomial.
\end{lemma}


\begin{proof} Let $F$ be the splitting field of the polynomial
$x^{(k-2)!}-1$ over $\textnormal{GF}(q)$. Clearly, $g(x)$ also
splits in $F$. Since the characteristic of $\textnormal{GF}(q)$ is
$p_0>k-2$, we know that $x^{(k-2)!}-1$ has no multiple roots in
$F$.

\verb| |If $g(x)=x^{j_0}$ with $0<j_0<m$, the conclusion is clear.
In other cases, write $g(x)=x^{j_0}f_1^{j_1}(x)\cdots
f_n^{j_n}(x)$, where  $n$ is the largest integer such that
$j_n>0$. By Lemma 4 and Lemma 5, $f_n(x)$ has a factor
$\Phi_n(x)$, which is relatively prime to $f_1(x)f_2(x)\cdots
f_{n-1}(x)$. It follows that each zero root of $\Phi_n(x)$ is a
root of $g(x)$ in $F$ of multiplicity $j_n<m$. Therefore, $g(x)$
can not be an $m$th power of a polynomial in
$\textnormal{GF}(q)[x]$.
\end{proof}

\begin{proof}[Proof of Theorem 1]
Let $\Omega=\{\beta|\beta\in{\textnormal{GF}}(q)\ \
{\textnormal{satisfies\ \ (2)}}\}$. It suffices to show that if
$q$ is large enough then $|\Omega|>0$.

Let $a=e^{\frac{2\pi i}{k(k-1)/2}}$ be a ${k(k-1)/2}$-th root of
unity.  For any integer $j$, we define $\psi(\theta^j)=a^j$. Since
$q-1\equiv k(k-1)(\bmod\ 2k(k-1))$, we know $\psi$ is a
multiplicative character of order $k(k-1)/2$ on
$\textnormal{GF}(q)$.

Let $f_1(x)$, $f_2(x)$, $\cdots$, $f_{k-2} (x)$ be the polynomials
given by (3). For each polynomial $f_j(x)$, consider the sum
$1+\psi(f_j(x))+\psi^2(f_j(x)) +\cdots+\psi^{k(k-1)/2-1}(f_j(x))$,
where $x\in\textnormal{GF}(q)$. By Lemma 3, we have

\begin{equation}1+\psi(f_j(x))+\cdots+\psi^{k(k-1)/2-1}(f_j(x))=\left\{
\begin{array}{ll}k(k-1)/2,&{\textnormal {if}}\ f_j(x)\in M,\\
1,&{\textnormal {if}}\ f_j(x)=0,\\
0,&{\textnormal{otherwise}}.\end{array}\right.
\end{equation}


Let $V(x)$ be a mapping from $\textnormal{GF}(q)$ to $\mathbb{C}$
defined by
\begin{equation}V(x)=[1-\psi(x)]\cdot
\prod_{j=2}^{\frac{k(k-1)}{2}-1}[\psi(x)-a^j], \ \ \forall
x\in\textnormal{GF}(q).
\end{equation}
Since $\psi$ is of order $k(k-1)/2$, we may write $V(x)$ as
follows,

\begin{equation}V(x)=c_0+c_1\psi(x)+c_2\psi^2(x)+\cdots+c_{\frac{k(k-1)}{2}-1}
\psi^{\frac{k(k-1)}{2}-1}(x),
\end{equation}
where the coefficients  $c_0, c_1,\cdots, c_{\frac{k(k-1)}{2}-1}$
are some complex numbers deduced from (5). It is easy to prove
that the coefficients $c_i$ have a common bound, i.e., each $c_i$
satisfies $|c_i|\leq2^{k(k-1)/2-1}$.

Notice that if $x\in M\theta^j$, then $\psi(x)=a^j$. Therefore,
from (5) we have
\begin{equation}V(x)=\left\{\begin{array}{ll}V(\theta),
&{\textnormal{if}\ \ }x\in
M\theta,\\
V(0),&{\textnormal{if}\ \ }x=0,\\
0,&{\textnormal{otherwise.}}
\end{array}\right.
\end{equation}
Write $b_0=V(\theta)$, $c_0=V(0)$. Clearly, we have
\begin{eqnarray}b_0&=&[1-a]\prod_{j=2}^{\frac{k(k-1)}{2}-1}[a-a^j]\not=0,\nonumber
\end{eqnarray}
\begin{eqnarray}c_0&=&(-1)^{k(k-1)/2}\prod_{j=2}^{\frac{k(k-1)}{2}-1}a^j\not=0.\nonumber\end{eqnarray}
The value of $c_0$ is of importance. In fact we have
$c_0=-a^{-1}$, which is deduced from the fact that $a$ is a
$k(k-1)/2$-th root of unity, i.e., there is an identity
$z^{k(k-1)/2}-1=(z-1)(z-a)(z-a^2)\cdots(z-a^{k(k-1)/2-1})$.

We now define a mapping  $H(x)$ from $\textnormal{GF}(q)$ to
$\mathbb{C}$ by
\begin{equation}H(x)=V(x)\prod_{j=1}^{k-2}[1+\psi(f_j(x))+\cdots+\psi^{\frac{k(k-1)}{2}-1}(f_j(x))],\
\forall x\in \textnormal{GF}(q).
\end{equation}
Roughly speaking, $H(x)$ is a {\em sieve}, that is, we can use
$H(x)$ to preserve the required elements $\beta\in\Omega$ and {\em
sift} most elements in the rest part
$\textnormal{GF}(q)\setminus\Omega$.

Now consider the sum
\begin{equation}\sum_{x\in{\scriptsize{\textnormal{GF}}}(q)}H(x)\ .\ \end{equation}

We partition $\textnormal{GF}(q)$ into three disjoint subsets,
\[\textnormal{GF}(q)=\Omega\dot{\cup}\Omega_1\dot{\cup}\Omega_2,\]
where $\Omega_1=\{\beta|\ \beta {\textnormal{\ is\ a\ root \ of \
}}f_1(x)f_2(x)\cdots f_{k-2}(x)=0\}$, and
$\Omega_2=\textnormal{GF}(q)\setminus(\Omega\cup\Omega_1)$. Note
that $f_j(x)$ has at most $j+1$ roots in $\textnormal{GF}(q)$, and
those polynomials have at least two  common zero roots (i.e., 0
and 1), hence we have $|\Omega_1|<1+2+\cdots+(k-2)<k^2$.

We have
\begin{equation}\sum_{x\in{\scriptsize{\textnormal{GF}}}(q)}H(x)=
\sum_{x\in\Omega}H(x)+\sum_{x\in\Omega_1}H(x)+
\sum_{x\in\Omega_2}H(x)\ .\end{equation} In view of (4) and (7),
we know that if $x\in\Omega$ then
$H(x)=b_0\cdot\left[\frac{k(k-1)}{2}\right]^{k-2}$,  while if
$x\in\Omega_2$ then $H(x)=0$. Therefore, we get
\begin{equation}\sum_{x\in{\scriptsize{\textnormal{GF}}}(q)}H(x)=b_0\cdot\left[\frac{k(k-1)}{2}\right]^{k-2}|
\Omega|+\sum_{x\in\Omega_1}H(x).\end{equation}

We substitute the expression of $V(x)$ in (6) into the
 right-hand side of (8), then  we expand out $H(x)$. For simplicity, we denote
$\psi(x)$ by $\psi$, $f_1(x)$ by $f_1$, and $\psi(f_1(x))$ by
$\psi(f_1)$, etc. If define $0^0=1$, then for every
$\alpha\in\textnormal{GF}(q)$ and  every integer $0\leq
i<k(k-1)/2$, we have  $\psi^i(\alpha)=\psi(\alpha^i)$. From (6)
and (8) we get

\begin{eqnarray}H(x)&=&c_0+\sum_{(j,j_1,j_2,\cdots,j_{k-2})}c_{j}\psi^{j}
\psi^{j_1}(f_1)\psi^{j_2}(f_2)\cdots\psi^{j_{k-2}}(f_{k-2})\nonumber\\
&=&c_0+\sum_{(j,j_1,\cdots,j_{k-2})}c_j\psi(x^jf_1^{j_1}f_2^{j_2}\cdots
f_{k-2}^{j_{k-2}}),\nonumber
\end{eqnarray}
where the indexes $j,j_1,j_2,\cdots,j_{k-2}$ are not all zero, and
each index belongs  to $\{0,$ $1,$ $\cdots,$
$\frac{k(k-1)}{2}-1\}$.


Then the sum  (9) becomes that
\begin{equation}\sum_{x\in{\scriptsize{\textnormal{GF}}}(q)}H(x)=\sum_{x\in\scriptsize{\textnormal{GF}}(q)}c_0+
\sum_{(j,j_1,\cdots,j_{k-2})}
\sum_{x\in\scriptsize{\textnormal{GF}}(q)}
c_j\psi(x^jf_1^{j_1}f_2^{j_2}\cdots f_{k-2}^{j_{k-2}}).
\end{equation}

Equating (11) and (12), we get that
\begin{eqnarray}\left[\frac{k(k-1)}{2}\right]^{k-2}b_0|\Omega|&=&c_0q
-\sum_{x\in\Omega_1}H(x)+\nonumber\\
&&+\sum_{(j,j_1,\cdots,j_{k-2})}
\sum_{x\in\scriptsize{\textnormal{GF}}(q)}
c_j\psi(x^jf_1^{j_1}f_2^{j_2}\cdots f_{k-2}^{j_{k-2}})\nonumber\\
&=&c_0q+S_1+S_2,
\end{eqnarray}
where $S_1=-\sum_{x\in\Omega_1}H(x)$, and
$S_2=\sum_{(j,j_1,\cdots,j_{k-2})}
\sum_{x\in\scriptsize{\textnormal{GF}}(q)}[\cdots]$.

Since $|\psi(x)|\leq1$, it follows from (8)  that
$|H(x)|\leq2^{k(k-1)/2}\left[\frac{k(k-1)}{2}\right]^{k-2}$. Hence
we have
\[\left|S_1\right|=\left|\sum_{x\in\Omega_1}H(x)\right|
\leq|\Omega_1||H(x)|\leq
k^22^{k(k-1)/2}\left[\frac{k(k-1)}{2}\right]^{k-2}.\]

By Lemma 7, we can use Weil's Theorem to conclude that
\[\left|\sum_{x\in\scriptsize{\textnormal{GF}}(q)}
c_j\psi(x^jf_1^{j_1}f_2^{j_2}\cdots f_{k-2}^{j_{k-2}})\right|\leq
|c_j|k^2\sqrt{q}\leq 2^{k(k-1)/2}k^2\sqrt{q},\]  then it follows
that
\[|S_2|=\left|\sum_{(j,j_1,j_2,\cdots,j_{k-2})}
\sum_{x\in\scriptsize{\textnormal{GF}}(q)}[\cdots]\right|\leq2^{k(k-1)/2}k^2\left[\frac{k(k-1)}{2}\right]^{k-1}\sqrt{q}.\]
Therefore, we find that

\begin{equation}|S_1|+|S_2|<2^{k(k-1)/2}k^2\left[\frac{k(k-1)}{2}\right]^{k-1}(\sqrt{q}+1).\end{equation}
From (13) and (14), we get that
\begin{eqnarray}\left[\frac{k(k-1)}{2}\right]^{k-2}|b_0||\Omega|&\geq&|c_0|q-(|S_1|+|S_2|)\nonumber\\
&>&|c_0|q-2^{k(k-1)/2}k^2\left[\frac{k(k-1)}{2}\right]^{k-1}(\sqrt{q}+1).\nonumber
\end{eqnarray}

Since $|c_0|=|-a^{-1}|=1$, the above inequality shows that if $q$
is large enough, say,
$\sqrt{q}>1+2^{k(k-1)/2}k^2\left[\frac{k(k-1)}{2}\right]^{k-1}$,
then $|\Omega|>0$,  which implies that there is an element
$\beta\in\textnormal{GF}(q)$ satisfying (2).

This completes the proof of Theorem 1.
\end{proof}

\begin{remark}A design $\mathcal{D}$ constructed via Lemma 2
has a regular block-transitive automorphism group
$G=\textnormal{GF}(q)^+:L$, but its full automorphism group
Aut$\mathcal {D}$ might be much bigger. For example, take $k=3$,
$q=7$, $M=\left<-1\right>$, and $L=\left<1\right>$ in
$\textnormal{GF}(7)$, then $B=\{0,1,3\}$ satisfies Lemma 2, but a
2-$(7,3,1)$ design is the projective plane of order $2$, and its
full automorphism group is isomorphic to $\textnormal{PSL}_2(7)$,
while $\textnormal{GF}(7)^{+}:L\cong Z_7$ is only corresponding to
the Singer cycles. Another example is the Netto system $N(q)$ when
$q\equiv7$ or $31(\bmod\ 36)$, which will be explained later. It
seems to be a difficult question to determine
$\textnormal{Aut}\mathcal{D}$ for the designs given by Lemma 2.
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Application to $k=3$}
In this section, we take $k=3$, $q\equiv7(\bmod\ 12)$. We apply
Lemma 2 and Theorem 1 to construct some 2-$(q,3,1)$ designs. We
define $G=\textnormal{GF}(q)^+:\left<\theta^6\right>$ and
$M=\left<\theta^3\right>$, where $\theta$ generates
$\textnormal{GF}(q)$.

For $k=3$, it is interesting to compare the Netto designs and the
designs constructed via Lemma 2. The Netto design $N(q)$  may be
explained as follows. For each prime power  $q\equiv 7(\bmod\
12)$, choose $\varepsilon$ to be a primitive sixth root of unity
in $\textnormal{GF}(q)$, then let $B=\{0,1,\varepsilon\}$. Let
$\textnormal {A}\Gamma^2\textnormal{L}(1,q)$ be the group of
permutations of $\textnormal{GF}(q)$ of the form $x\rightarrow
\alpha^2x^{\tau}+\beta $, where
$\alpha,\beta\in\textnormal{GF}(q)$, $\alpha\not=0$ and $\tau$ is
a field automorphism. To define the Netto system $N(q)$, let the
points be the elements of $\textnormal{GF}(q)$, and let  the
blocks be the images of $B$ under the action of $\textnormal
{A}\Gamma^2\textnormal{L}(1,q)$. Except for $q=7$ the full
automorphism of $N(q)$ is
$\textnormal{A}\Gamma^2\textnormal{L}(1,q)$.

In \cite{gran}, M. Grannell, T. Griggs and J. Murphy constructed
some 2-$(q,3,1)$ designs, which may be seen as  the special case
of Lemma 2 for $k=3$. The notation they used is different from
ours.  We state the fundamental theory upon which their
computations are based in the following theorem.

\begin{theorem} \textnormal{({\cite{gran}})}\  \ Let $p$ be a prime
with $p\equiv7(\bmod\ 12)$. If there is an element
$\beta\in\textnormal{GF}(p)$ such that $\{M\beta,
M(1-\beta)\}=\{M\theta,\ M\theta^2\}$, then
$\mathcal{D}=(\textnormal{GF}(p),B^G)$ is a 2-$(p,3,1)$ design,
where $B=\{0,1,\beta\}$.
\end{theorem}

M.Grannell et al. pointed out that up to $p\leq 75079$, the
required  elements  $\beta $ in Theorem 9 exist. In the next
theorem we  prove the existence of such an element $\beta$ for
each prime power $q\equiv7(\bmod\ 12)$, and compare those designs
with the Netto designs.

\begin{theorem} Let $q$ be a prime power with $q\equiv7(\bmod\
12)$. Then
\begin{description}\item{\textnormal{(1)}}\ \ If $q\equiv7$
 or 31$(\bmod\ 36)$,
then the class of 2-$(q,3,1)$ designs constructed via Theorem 9
contains the Netto designs;

\item{\textnormal{(2)}}\ \ If $q>7$, then there are at least three
pairwise distinct elements $\beta_1$, $\beta_2$, $\beta_3$ of
$\textnormal{GF}(q)$, which means that at least one $\beta_i$ is
not a primitive sixth root of unity, such that $B=\{0,1,\beta_i\}$
\textnormal{($i=1,2,3$)} satisfies Theorem 9.
\end{description}
\end{theorem}

\begin{proof} (1) Consider the block $B=\{0,1,\varepsilon\}$ of $N(q)$. By
the definition of $\varepsilon$, we have
$\varepsilon=\theta^{\pm(q-1)/6}$, and
$\varepsilon^2-\varepsilon+1=0$ holds, hence $\triangle
B=\{1,\varepsilon,\varepsilon^2\}$. It follows that if
$\varepsilon\not\in M$, which is equivalent to
$3\nmid\frac{q-1}{6}$, or $q\equiv7$ or 31$(\bmod\ 36)$, then
$\triangle B$ is a system of the representatives of the cosets of
$M$ in $\textnormal{GF}(q)^\times$. While if $\varepsilon\in M$,
i.e., $q\equiv 19(\bmod\ 36)$, then $\triangle B\subseteq M$.

(2) The proof is similar to the proof of Theorem 1. We only give
some  points.

Select an element $\beta\in \textnormal{GF}(q)$ satisfying that
\begin{equation}\beta\in M\theta\cup M\theta^{-1},\ \ \  \textnormal{and }\ \
  \beta(\beta-1)\in M.\end{equation}
Then $B=\{0,1,\beta\}$ satisfies Theorem 9. Let $\Omega$ denote
the set of such elements $\beta$. To prove (2) is to prove that
$|\Omega|>2$ for $q>7$.

Now partition $\textnormal{GF}(q)$ into three disjoint parts and
denote the partition by
$\textnormal{GF}(q)=\Omega\cup\{0,1\}\cup\Omega_1$, where
$\Omega_1=\textnormal{GF}(q)\setminus(\Omega\cup\ \{0,1\})$.

Let $\psi$ be a multiplicative character of order 3 on
$\textnormal{GF}(q)$ and let $f(x)=x(x-1)$.  Let $H(x)$ be a
mapping from $\textnormal{GF}(q)$ to $\mathbb{C}$ defined by
\[H(x)=[2-\psi(x)-\psi^{-1}(x)][1+\psi(f(x))+\psi^{-1}(f(x))],\ \ \forall x\in\textnormal{GF}(q).\]
Then we have
\begin{eqnarray}
H(x)&=&2-\psi(x)-\psi^{-1}(x)+2\psi(f(x))+2\psi^{-1}(f(x))\nonumber\\
&&-\psi(xf(x))-\psi(xf^2(x))-\psi(x^2f(x))-\psi(x^2f^2(x)).
\end{eqnarray}

Now consider $\sum_{x\in{\scriptsize{\textnormal{GF}}(q)}}H(x)$.
It can be verified directly that $H(0)=2$, and  if $x\in\Omega$
then $H(x)=9$. While if $x=1$ or $x\in\Omega_1$, then $H(x)=0$.
Thus from the left-hand side of (16) we get

\[\sum_{x\in{\scriptsize{\textnormal{GF}}(q)}}H(x)=9|\Omega|+2.\]
We use Weil's Theorem to estimate the sums such as
$\sum_{x\in{\scriptsize{\textnormal{GF}}(q)}}\psi(xf(x))$ and \linebreak
$\sum_{x\in{\scriptsize{\textnormal{GF}}(q)}}\Psi(x^2f(x))$,
etc. Then from the right-hand side of (16), we get

\[\sum_{x\in{\scriptsize{\textnormal{GF}}(q)}}H(x)=2q+\cdots
\geq2q-8\sqrt{q}.\]
Therefore we get
$9|\Omega|+2\geq2q-8\sqrt{q}$. If $q\geq 40$, then
$|\Omega|\geq3$, as required.

In the remaining case where $q=19$ or 31, we can find the required
elements $\beta$ directly. When $q=19$, choose 2 to be a
generating element of $\textnormal{GF}(19)$. We have
$\varepsilon\in\{8,-7\}$, $M=\left<2^3\right>$. Then one may
verify that $B=\{0,1,16\}$ satisfies the requirement.

For $\textnormal{GF}(31)$, choose 3 to be a generating element. We
have $\varepsilon\in\{-5,6\}$, $M=\left<3^3\right>$. Then
$B=\{0,1,12\}$  is such a subset.
\end{proof}


\vspace{2mm}

\begin{remark} As pointed out in \cite{gran}, if $\beta$ satisfies (15),
then the designs generated by the blocks $B=\{0,1,\beta\}$ where
$\beta\in\{\beta, 1-\beta, 1/(1-\beta), \beta/(\beta-1),
1-1/\beta, 1/\beta\}$ are all isomorphic.  We believe that the
designs generated by $(0,1,\beta)$, where $\beta$ is not a
primitive sixth root of unity, are not isomorphic to the  Netto
designs. However  we are unable to prove this.
\end{remark}

The 2-$(v,3,1)$ designs having a block transitive automorphism
group have been classified by P. C. Clapham~\cite{clap}. Here we
quote Clapham's result from \cite{cam2}.

\noindent
\begin{theorem}[Clapham's Theorem]
Let $K$ act as a block transitive group on a 2-$(v,3,1)$ design
$\mathcal {D}$. Then one of the following holds

\begin{description}\item{\textnormal{(1)}}\ $K$ acts 2-transitively on points,
\item{\textnormal{(2)}}\ $K$ has odd order and is a subgroup of
the {\textnormal{A}\textnormal{$\Gamma$L}}$(1,p^d)$ containing the
translation subgroup, where $p$ is a prime and $d$ is a natural
number and one of the following holds:
\begin{description}\item{{\textnormal{(2a)}}}\ $\mathcal D $ is an
affine geometry of dimension $d$ over $\textnormal{GF}(3)$, $d$ is
odd and $K$ has rank 2 on points, \item{\textnormal{(2b)}}
$\mathcal{D}$ is a Netto design, \item{\textnormal{(2c)}}\ $K$ has
rank 7 on the points and $p^d\equiv7(\bmod\ 12)$.
\end{description}
\end{description}

\end{theorem}
There has been a complete classification of those designs in (1)
of Theorem 12. Note that
$G=\textnormal{GF}(q)^+:\left<\theta^6\right>$ is a subgroup of
$\textnormal {A}\Gamma^2\textnormal{L}(1,q)$, and the setwise
stabilizer of $\{0,1,\varepsilon\}$ in $G$ is trivial, thus  the
Netto designs $N(q)$ together with $G$ are examples of (2c) of
Theorem 12. Other examples also exist by Theorem 10.

\subsection*{Acknowledgements}
The authors would like to thank the referee for helpful
suggestions.




























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