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 \title{\bf Generalized Balanced Tournament
 Designs with Block Size Four\thanks{Research of Y. M. Chee, H. M. Kiah, and C. Wang is supported in part
 by the Singapore National
 Research Foundation under Research Grant NRF-CRP2-2007-03. Research
 of C. Wang is also supported in part by NSFC under Grants No.
 11271280 and 10801064.}}

% input author, affilliation, address and support information as follows;
% the address should include the country, and does not have to include
% the street address

\author{Yeow Meng Chee\qquad Han Mao Kiah \\
\small School of Physical and Mathematical Sciences \\[-0.8ex]
\small Nanyang Technological University\\[-0.8ex]
\small Singapore 637371\\
\small\tt \{YMChee,KIAH0001\}@ntu.edu.sg\\
\and
Chengmin Wang \thanks{Corresponding author.} \\
\small School of Science\\[-0.8ex]
\small Jiangnan University\\[-0.8ex]
\small Wuxi, China 214122\\
\small\tt wcm@jiangnan.edu.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{XXX}{XXX}\\
\small Mathematics Subject Classifications: 05B05, 94B25}

\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
% should be informative, clear, and as complete as possible. Phrases
% like "we investigate..." or "we study..." should be kept to a minimum
% in favor of "we prove that..."  or "we show that...".  Do not
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 \begin{abstract}
 In this paper, we remove the outstanding values $m$
 for which the existence of a GBTD$(4,m)$ has not been decided
 previously. This leads to a complete solution to the existence
 problem regarding GBTD$(4,m)$s.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} generalized balanced tournament design; holey generalized balanced tournament
  design; starter-adder
\end{abstract}

\section{Introduction}
 A {\em set system} is a pair ${\frak S}=(X,{\cal B})$, where $X$ is a finite set
 of {\em points} and  ${\cal B}$ is a collection of subsets of $X$.
 Elements of $\cal B$ are called {\em blocks}.
 The {\em order} of $\frak S$ is $|X|$, the number of points.
 Let $K$ be a set of positive integers.
 A set system $(X,{\cal B})$ is said to be {\em $K$-uniform} if $|B|\in K$ for all $B\in {\cal
 B}$.
 Let $(X,{\cal B})$ be a set system and $S\subseteq X$. A {\em partial $\alpha$-parallel class}
 over $X\backslash S$ of $(X,{\cal B})$ is a set of blocks ${\cal A}\subseteq{\cal B}$ such that
 each point of $X\backslash S$ occurs in exactly $\alpha$ blocks of ${\cal A}$,
 and each point of $S$ occurs in no block of ${\cal A}$. A {\em partial $\alpha$-parallel class}
 over $X$ is simply called an {\em $\alpha$-parallel class}.
 We adopt the convention that if $\alpha$ is not specified, then it is taken to be one, so that
a {\em parallel class} refers to a $1$-parallel class. A set system
$(X,{\cal B})$ is said to be {\em resolvable} if $\cal B$ can be
partitioned into parallel classes.


A {\em balanced incomplete block design of order $v$, block size
$k$, and index $\lambda$}, denoted by $(v,k,\lambda)$-BIBD, is a
$\{k\}$-uniform set system $(X,{\cal B})$ of order $v$ such that
every $2$-subset of $X$ is contained in precisely $\lambda$ blocks
of $\cal B$.
 A resolvable $(km, k, k - 1)$-BIBD $(X,{\cal B})$ is called a {\em generalized balanced tournament
 design} (GBTD), or simply a GBTD$(k,m)$, if the $m(km-1)$ blocks
 of ${\cal B}$ are arranged in an $m\times (km - 1)$ array such that
\begin{enumerate}[(i)]
 \item the set of blocks in each column is a parallel class, and
 \item each point of $X$ is contained in at most $k$ cells of each row.
 \end{enumerate}
GBTDs were introduced by Lamken \cite{Lamken1} and are useful in the
construction of many combinatorial designs, including resolvable,
 near-resolvable, doubly resolvable, and doubly near-resolvable
 balanced incomplete block designs (see \cite{CD, Lamken1}).
 More recently, GBTDs have also found applications in near constant-composition codes \cite{YYW}, and codes
 for power line communications \cite{CKLW}.

%Concerning the existence of GBTDs, the following is known.
Schellenberg et al. \cite{SVV} showed that a $\text{GBTD}(2,m)$
exists for all positive
 integers $m\neq  2$. Lamken \cite{Lamken2}
showed that a $\text{GBTD}(3,m)$ exists for all positive
 integers $m\neq  2$.
 %For $k\geq 4$, complete results on the existence of $\text{GBTD}(k,m)$ are not known.
 For $k=4$, Yin et al. \cite{YYW} obtained the following.

 \begin{theorem}[Yin et al. \cite{YYW}]
 \label{old GBTD}
 A $\text{\rm GBTD}(4,m)$ exists
 for all positive integers $m\geq 5$, except
 possibly when $m\in\{28, 32, 33, 34, 37, 38, 39, 44\}$.
 \end{theorem}

 The purpose of this paper is to remove all the remaining eight possible exceptions in
 Theorem \ref{old GBTD} and to show that a $\text{GBTD}(4,m)$ exists for $m=4$ but not
 for $m\in\{2,3\}$.

 \begin{theorem}\label{main th}
 For each $m \in\{4,28,32,33,34,37,38,39,44\}$, a GBTD$(4,m)$ exists.
 For $m=2$ and $3$, a GBTD$(4,m)$ does not exist.
 \end{theorem}

 A $\text{GBTD}(4,1)$ exists trivially. Combining Theorem \ref{old GBTD} and Theorem \ref{main th}, the existence of $\text{GBTD}(4,m)$ is now
 completely determined.

 \begin{theorem}\label{GBTD(4,m)}
 A $\text{\rm GBTD}(4,m)$ exists if and only if $m\geq 1$ and $m\neq 2, 3$.
 \end{theorem}


 %We outline our proof strategy. For $m\in\{3,4\}$, existence of
% $\text{GBTD}(4,m)$ is determined by computer search. To deal with the cases
% $m\in\{28, 32, 33, 34, 37, 38, 39, 44\}$,
% we introduce a new combinatorial design, an {\em incomplete holey GBTD}, that
% extends the concept of {\em holey GBTD} introduced by Yin et al. \cite{YYW}, to allow
% the appearance of an empty subarray.
% This increased flexibility allows the existence of
% a larger class of designs whose empty subarray can be filled to produce the required GBTDs.
% The remaining case of $m=2$ is handled by the following general nonexistence result.

 We first present a non-existence result.
 \begin{prop}\label{prop:2}
A $\text{\rm GBTD}(k,2)$ does not exist for all $k\geq 2$.
\end{prop}

\noindent {\bf Proof:} Suppose $(X,{\cal B})$ is a $(2k,k,k-1)$-BIBD
whose blocks are organized into a $2\times (2k-1)$ array to form a
$\text{GBTD}(k,2)$. Since $(X,{\cal B})$ is a $(2k,k,k-1)$-BIBD,
each point in $X$ appears in exactly $2k-1$ blocks, and hence each
point must appear either $k$ times in the first row and $k-1$ times
in the second row, or vice versa.
%Let $n_r$ denote the number of points in $X$ that appears $r$ times in the first row.
%Then we have
%\begin{eqnarray*}
%n_{k-1}+n_{k} &=& 2k, \\
%(k-1)n_{k-1}+kn_k &=& k(2k-1).
%\end{eqnarray*}
%This gives $n_{k-1}=n_k=k$.

Consider a point $x\in X$ that appears $k$ times in the first row
and $k-1$ times in the second row. Let $y\in X$ be distinct from
$x$. The cells in the first row can be classified as follows:
\begin{enumerate}[(i)]
\item Type-$xy$: a cell that contains both $x$ and $y$.
\item Type-$x\bar{y}$: a cell that contains $x$ but not $y$.
\item Type-$\bar{x}y$: a cell that contains $y$ but not $x$.
\item Type-$\bar{x}\bar{y}$: a cell that contains neither $x$ nor $y$.
\end{enumerate}
Let $\alpha$ and $\beta$ be the number of type-$xy$ cells and
type-$\bar{x}y$ cells in the first row, respecitvely. Then we have
the table
\begin{center}
T1=
\begin{tabular}{|r|c|c|c|c|}
\hline
& Type-$xy$ & Type-$x\bar{y}$ & Type-$\bar{x}y$ & Type-$\bar{x}\bar{y}$ \\
\hline
\# cells in first row & $\alpha$ & $k-\alpha$ & $\beta$ & $k-1-\beta$  \\
\hline
\# cells in second row & $k-1-\beta$ & $\beta$ & $k-\alpha$ & $\alpha$  \\
\hline
\end{tabular} ,
\end{center}
where the second row is obtained from the first by the following
observation: if a cell is of type-$xy$ (respectively,
type-$x\bar{y}$, type-$\bar{x}y$, type-$\bar{x}\bar{y}$) in the
first row, then the cell in the second row of the corresponding
column is of type-$\bar{x}\bar{y}$ (respectively, type-$\bar{x}y$,
type-$x\bar{y}$, type-$xy$). On the other hand, the total number of
type-$xy$ cells is $k-1$, since $(X,{\cal B})$ is a BIBD of index
$k-1$. Hence, we have $\alpha+(k-1-\beta)=k-1$, implying
$\alpha=\beta$.

%Table T1 can therefore be revised to
%\begin{center}
%T2=
%\begin{tabular}{|r|c|c|c|c|}
%\hline
%& Type-$xy$ & Type-$x\setminus y$ & Type-$y\setminus x$ & Type-$\setminus xy$ \\
%\hline
%\# cells in first row & $\alpha$ & $k-\alpha$ & $\alpha$ & $k-1-\alpha$  \\
%\hline
%\# cells in second row & $k-1-\alpha$ & $\alpha$ & $k-\alpha$ & $\alpha$  \\
%\hline
%\end{tabular}
%\end{center}

Considering the number of occurrences of $y$ in the first row and
the number of occurrences of $y$ in the second row of the GBTD give
the inequalities
\begin{eqnarray*}
\alpha+\beta & \leq& k, \\
2k-1-\alpha-\beta & \leq& k,
\end{eqnarray*}
from which, and $\alpha=\beta$ shown earlier, follow that
\begin{equation*}
\alpha=\lfloor k/2\rfloor.
\end{equation*}
%\begin{equation*}
%\alpha=\begin{cases}
%k/2,&\text{if $k$ is even;} \\
%(k-1)/2,&\text{if $k$ is odd.}
%\end{cases}
%\end{equation*}
Table T1 can therefore be revised to
\begin{center}
T2=
\begin{tabular}{|r|c|c|c|c|}
\hline
& Type-$xy$ & Type-$x\bar{y}$ & Type-$\bar{x}y$ & Type-$\bar{x}\bar{y}$ \\
\hline
\# cells in first row & $\lfloor k/2\rfloor$ & $\lceil k/2\rceil$ & $\lfloor k/2\rfloor$ & $\lceil k/2\rceil-1$  \\
\hline
\# cells in second row & $\lceil k/2\rceil-1$ & $\lfloor k/2\rfloor$ & $\lceil k/2\rceil$ & $\lfloor k/2\rfloor$  \\
\hline
\end{tabular} .
\end{center}
%\begin{center}
%T3=
%\begin{tabular}{|r|c|c|c|c|}
%\hline
%$k$ even & Type-$xy$ & Type-$x\setminus y$ & Type-$y\setminus x$ & Type-$\setminus xy$ \\
%\hline
%\# cells in first row & $k/2$ & $k/2$ & $k/2$ & $k/2-1$  \\
%\hline
%\# cells in second row & $k/2-1$ & $k/2$ & $k/2$ & $k/2$  \\
%\hline
%\end{tabular}
%\end{center}
%and
%\begin{center}
%T3$'$=
%\begin{tabular}{|r|c|c|c|c|}
%\hline
%$k$ odd & Type-$xy$ & Type-$x\setminus y$ & Type-$y\setminus x$ & Type-$\setminus xy$ \\
%\hline
%\# cells in first row & $(k-1)/2$ & $(k-1)/2+1$ & $(k-1)/2$ & $(k-1)/2$  \\
%\hline
%\# cells in second row & $(k-1)/2$ & $(k-1)/2$ & $(k-1)/2+1$ & $(k-1)/2$  \\
%\hline
%\end{tabular}
%\end{center}


Counting in two ways the number of elements in the set
\begin{equation*}
\{(y,C): \text{$y\in X$, $y\not=x$, and $C$ is a cell of type-$xy$
in the second row}\}.
\end{equation*}
gives
\begin{equation*}
(2k-1)(\lceil k/2\rceil-1)=(k-1)^2,
\end{equation*}
which is a contradiction.
%This point $y$ appears
%in exactly $k-1$ cells with $x$, since $(X,{\cal B})$ is a BIBD with index $k-1$, and
%in exactly $k$ cells without $x$.
%Let $\alpha$ be the number of cells in the first row containing both $x$ and $y$,
%and $\beta$ be the number of cells in the first row containing $y$ but not $x$.
%Hence,
%there are $k-1-\alpha$ cells in the second row containing both $x$ and $y$, and
%$k-\beta$ cells in the second row containing $y$ but not $x$.
%The $k-\alpha$ columns in the first row that contain $x$ but not $y$ must
%contain $y$ but not $x$ in the second row of the corresponding columns. Hence, the
%frequency of appearance of $y$ in the second row is at least
%$(k-\alpha)+(k-1-\alpha)=2k-1-2\alpha$. Since $y$ can appear at most $k$ times
%in the second row, we have $\alpha\geq \lceil (k-1)/2\rceil=k/2$. It follows that $y$ can appear
%with $x$ in at most $k/2-1$ cells of the second row.
%
%Now, count in two ways the number of pairs $(y,C)$ such that $y\in X$, $y\not=x$, and
%$C$ is a cell in the second row containing both $x$ and $y$. Clearly, since each of the
%$k-1$ cells $C$
%in the second row containing $x$ also contains $k-1$ other points, the number of such
%pairs $(y,C)$ is precisely $(k-1)^2$. On the other hand, Since $y$ can appear with
%$x$ in at most $(k-1)/2$ cells of the second row, the number of such pairs $(y,C)$ is
%at most $(2k-1)(k-1)/2=(2k^2-3k+1)/2$.
\hfill $\Box$

 \section{Existence of GBTD$(4,m)$s with $m=3$ and 4}

 For a positive integer $n$, the set $\{1,2,\ldots,n\}$ is denoted by $[n]$.
 Let $\Sigma$ be a set of $q$ symbols. A {\em $q$-ary code of length $n$} over $\Sigma$
 is a subset ${\cal C}\subseteq\Sigma^n$. Elements of $\cal C$ are called {\em codewords}.
 The {\em size} of $\cal C$ is the number of codewords in $\cal C$. For $i\in[n]$, the $i$th
 coordinate of a codeword ${\sf u}\in{\cal C}$ is denoted ${\sf u}_i$, so that
 ${\sf u}=({\sf u}_1,{\sf u}_2,\ldots,{\sf u}_n)$.

 The {\em symbol weight} of ${\sf u}\in\Sigma^n$, denoted $\text{swt}({\sf u})$,
 is the maximum frequency of appearance of a symbol in $\sf u$, that is,
 \begin{equation*}
 \text{swt}({\sf u}) = \max_{\sigma\in\Sigma} | \{ {\sf u}_i=\sigma:i\in[n]\}|.
 \end{equation*}
 A code has {\em constant symbol weight $w$} if all of its codewords have symbol weight
 exactly $w$.
 The {\em (Hamming) distance} between ${\sf u},{\sf v}\in\Sigma^n$ is
 $d_\text{H}({\sf u},{\sf v})=|\{ i\in[n]: {\sf u}_i={\sf v}_i\}|$, the number
 of coordinates at which $\sf u$ and $\sf v$ differ.
 A code ${\cal C}$ is said to have {\em distance}
 $d$ if $d_\text{H}({\sf u},{\sf v})\geq d$ for all distinct ${\sf u},{\sf v}\in{\cal C}$.
 A $q$-ary code of length $n$, constant symbol weight $w$, and distance $d$ is referred to as an
 {\em $(n,d,w)_q$-symbol weight code}. An $(n, d, w)_q$-symbol weight code with maximum size is said to be {\em optimal}.

 Chee et al. \cite{CKLW} established the following
 relation between a GBTD and a symbol weight code.

 %In a $\text{GBTD}(k,m)$, given any point $x$ and any column $j$, there is a unique row
% that contains the point $x$ in column $j$. Hence, for each point $x\in X$ of a
% $\text{GBTD}(k,m)$ $(X,{\cal B})$, we may correspond the codeword
% ${\sf c}(x)=(r_1,r_2,\ldots,r_{km-1})\in[m]^{km-1}$, where $r_j$ is the row in which point $x$
% appears in column $j$. It is obvious that ${\cal C}=\{{\sf c}(x):x\in X\}$is an $m$-ary code
% of length $km-1$ over the alphabet $[m]$. This correspondence is precisely the one used
% by Semakov and Zinoviev {\cite{SemakovZinoviev:1968}} to show the equivalence between
% equidistant codes and resolvable balanced incomplete block designs.
%
% For distinct points $x,y\in X$, the distance between ${\sf c}(x)$ and ${\sf c}(y)$ is the number
% of columns for which $x$ and $y$ are not both contained in the same row. Since there are
% exactly $k-1$ blocks containing both $x$ and $y$, and that no two such blocks can occur in the
% same column of the $\text{GBTD}(k,m)$, the distance between ${\sf c}(x)$ and ${\sf c}(y)$
% is $(km-1)-(k-1)=k(m-1)$. This distance is independent of $x$ and $y$, making
% $\cal C$ an {\em equidistant code}, that is, the distance between any two codewords
% of $\cal C$ is $k(m-1)$.
%
% Next, we determine the symbol weight ${\sf c}(x)$, for $x\in X$. From the construction of
% ${\sf c}(x)$, the number of times a symbol $i$ appears in ${\sf c}(x)$ is the number of cells in
% row $i$ that contains $x$. By the definition of a $\text{GBTD}(k,m)$, this number is at most $k$.
% However, $\text{swt}({\sf u})\geq \lceil n/q\rceil$ for any ${\sf u}\in\Sigma^n$, so that
% $\text{swt}({\sf c}(x))\geq \lceil (km-1)/m\rceil=k$. It follows that $\cal C$ has constant symbol
% weight $k$.
%
% This construction of a symbol weight code from a $\text{GBTD}$ can be easily reversed, so that
% we have the following result.

 \begin{theorem}[Chee et al. \cite{CKLW}]
 \label{GBTDSWC}
 A $\text{\rm GBTD}(k,m)$ exists if and only if an optimal
 $(km-1,k(m-1),k)_m$-symbol weight code exists.
 % $m$-ary code of length $km-1$, constant symbol weight $k$, and size $km$,
% that is of equidistance $k(m-1)$.
 \end{theorem}

 In view of Theorem \ref{GBTDSWC}, to prove the nonexistence of a
 $\text{GBTD}(4,3)$, it suffices to show that there does not exist a ternary code of length
 $11$, constant symbol weight four, and size $12$, that is of equidistance eight.
 Consider the {\em Gilbert graph} $G=(V,E)$, where $V=\{{\sf u}\in [3]^{11}:\text{swt}({\sf u})=4\}$
 and two vertices
 ${\sf u},{\sf v}\in V$ are adjacent in $G$ if and only if $d_\text{H}({\sf u},{\sf v})=8$. Then
 there exists a ternary code of length
 $11$, constant symbol weight four, and size $12$, that is of equidistance eight if and
 only if there exists a clique of size 12 in $G$. It is not hard to see that $G$ is vertex-transitive
 so that we can just search for a clique of size 11 in $G'$, the subgraph of $G$ induced by
 the set of vertices $\{{\sf v}\in V: d_\text{H}({\sf u},{\sf v})=8\}$ for some fixed ${\sf u}\in V$.
 This induced subgraph $G'$ has 8001 vertices and 7233060 edges. We solve this clique-finding
 problem using {\tt Cliquer}, an implementation of \"{O}sterg{\aa}rd's
clique-finding algorithm by Niskanen and \"{O}sterg{\aa}rd
{\cite{NiskanenOstergard:2003}}.
 The result is that the largest clique in $G'$ has size 10. Consequently, we have the following.

 \begin{prop}\label{prop:3}
 There does not exist a $\text{\rm GBTD}(4,3)$.
 \end{prop}

 There exists, however, a $\text{GBTD}(4,4)$. Unfortunately, a $\text{GBTD}(4,4)$
 is too large to be found by the method of clique-finding above. Instead, to curb the
 search space, we assume
 the existence of some automorphisms acting on the $\text{GBTD}(4,4)$ to be found.
 Let $(X,{\cal B})$ be a $\text{GBTD}(4,4)$, where $X={\mathbb Z}_4\times{\mathbb Z}_4$.
 If $B\subseteq X$ and $x\in X$, $B+x$ denotes the set $\{b+x: b\in B\}$. If $\sf A$ is
 an array over $X$ and $x\in X$, ${\sf A}+x$ denotes the array obtained by adding $x$
 to every entry of $\sf A$.
 For succinctness, a point $(x,y)\in{\mathbb Z}_4\times{\mathbb Z}_4$ is
 sometimes written $xy$.

 The $\text{GBTD}(4,4)$ we construct contains the $4\times 3$ subarray
\begin{equation*}
\renewcommand{\arraystretch}{1.2}
{\sf A}_0=
 \begin{array}{|ccc|}
 \hline
 \{00,  02,  20,  22\}&\{11,  13,  31,  33\}&\{10,  12,  30,  32\}\\
 \{01,  03,  21,  23\}&\{00,  02,  20,  22\}&\{11,  13,  31,  33\}\\
 \{10,  12,  30,  32\}&\{01,  03,  21,  23\}&\{00,  02,  20,  22\}\\
 \{11,  13,  31,  33\}&\{10,  12,  30,  32\}&\{01,  03,  21,  23\}\\
 \hline
 \end{array} .
\end{equation*}
The blocks in ${\sf A}_0$ contain exactly the 2-subsets
$\{ab,cd\}\subseteq X$, where $a+c\equiv b+d\equiv 0\bmod 2$, each
thrice. The remaining $4\times 12$ subarray of the
$\text{GBTD}(4,4)$ is built from a set of 12 base blocks ${\cal
S}=\{B_{i,j}:\text{$i\in [3]$ and $0\leq j\leq 3$}\}$ as follows.
Let
\begin{equation*}
{\sf A}_1=
\begin{array}{|ccc|}
\hline
B_{1,0} & B_{2,0} & B_{3,0} \\
B_{1,1} & B_{2,1} & B_{3,1} \\
B_{1,2} & B_{2,2} & B_{3,2} \\
B_{1,3} & B_{2,3} & B_{3,3} \\
\hline
\end{array} .
\end{equation*}
Then the $4\times 12$ subarray is given by
\begin{equation*}
\begin{array}{|cccc|}
\hline
{\sf A}_1 & {\sf A}_1+(0,1) & {\sf A}_1+(0,2) & {\sf A}_1+(0,3) \\
\hline
\end{array} .
\end{equation*}
For
\begin{equation*}
\begin{array}{|ccccc|}
\hline
{\sf A}_0 & {\sf A}_1 & {\sf A}_1+(0,1) & {\sf A}_1+(0,2) & {\sf A}_1+(0,3) \\
\hline
\end{array}
\end{equation*}
to be a $\text{GBTD}(4,4)$, the following conditions are imposed:
\begin{enumerate}[(i)]
\item $\bigcup_{j=0}^3 B_{i,j}={\mathbb Z}_4\times{\mathbb Z}_4$, for $i\in[3]$, so that every
column is a parallel class.
\item For each $j$, $0\leq j\leq 3$, each element of ${\mathbb Z}_4$
appears exactly three times as a first coordinate among the elements
of $\bigcup_{i=1}^3 B_{i,j}$, so that every row contains each
element of ${\mathbb Z}_4\times{\mathbb Z}_4$ at most three times.
\item Let $k,l\in{\mathbb Z}_4$ and define $\Delta_{k,l}{\cal S}$ to be the multiset
$\bigcup_{A\in{\cal S}}\{x-y:(k,x),(l,y)\in A\}$. Then
\begin{equation*}
\Delta_{k,l}{\cal S}=\begin{cases}
\{1,1,1,3,3,3\},&\text{if $k=l$ or $k+l\equiv0\bmod 2$}; \\
\{0,0,0,1,1,1,2,2,2,3,3,3\},&\text{otherwise}.
\end{cases}
\end{equation*}
This ensures that every $2$-subset of $X$ not contained in any block
in ${\sf A}_0$ is contained in exactly three blocks in ${\sf A}_1$,
${\sf A}_1+(0,1)$, ${\sf A}_1+(0,2)$, or ${\sf A}_1+(0,3)$.
\end{enumerate}
A computer search found the following array ${\sf A}_1$ that
satisfies all the conditions above.
\begin{equation*}
{\sf A}_1=
\begin{array}{|ccc|}
\hline
\{23,22,32,11\} & \{10,00,21,11\} & \{00,01,30,33\} \\
\{20,01,30,33\} & \{33,02,03,12\} & \{10,13,22,23\} \\
\{31,00,12,21\} & \{01,13,20,32\} & \{02,11,21,32\} \\
\{02,10,13,03\} & \{22,23,30,31\} & \{03,12,20,31\} \\
\hline
\end{array} .
\end{equation*}
Consequently, we have the following.
 \begin{prop}\label{prop:4}
 There exists a $\text{\rm GBTD}(4,4)$.
 \end{prop}

 %A {\em group divisible design} of block size $k$ and index $\lambda$, or a $(k,\lambda)$-GDD,
% is a triple $(X, {\cal G}, {\cal B})$ where $X$ is a finite set of ({\em points}),
% ${\cal G}$ is a partition of $X$ into subsets (called {\em groups}), and
% ${\cal B}$ is a set of subsets of size $k$ (called {\em blocks}) of $X$, such that every pair of
% points from distinct groups occurs in exactly $\lambda$ blocks,
% and any pair of points from the same group occur in no block.
% The type of the GDD is defined to be the multiset $T
% = \{|G| : G \in {\cal G}\}$, which is usually denoted by an ``exponential"
% notation: a type $1^i2^j3^k\ldots$ denotes $i$ occurrences of 1,
% $j$ occurrences of 2, {\em etc}.
%
% A set of blocks of a GDD $(X, {\cal G}, {\cal B})$ is called a {\em partial $\alpha$-parallel
% class} over $X\backslash S$ where $S\subseteq X$ if each point of $X\backslash S$ occurs in exactly $\alpha$ blocks,
% while any point of $S$ occurs in no block. If $S =\emptyset$, it is called an
% {\em $\alpha$-parallel class} over $X$. Whenever $\alpha = 1$,
% we simply say a (partial) parallel class, instead of a (partial)
% $\alpha$-parallel class.
 %A GDD is termed {\em resolvable} if its blocks can be
% partitioned into parallel classes.

\section{Incomplete Holey GBTDs}

Let $(X,{\cal B})$ be a set system, and let ${\cal G}$ be a
partition of $X$ into subsets, called {\em groups}. The triple
$(X,{\cal G},{\cal B})$ is a {\em group divisible design} (GDD) of
index $\lambda$ when every $2$-subset of $X$ not contained in a
group appears in exactly $\lambda$ blocks, and $|B\cap G|\leq 1$ for
all $B\in{\cal B}$ and $G\in{\cal G}$. We denote a GDD $(X,{\cal
G},{\cal B})$ of index $\lambda$ by $(K,\lambda)$-GDD if $(X,{\cal
B})$ is $K$-uniform. The {\em type} of a GDD $(X,{\cal G},{\cal B})$
is the multiset $[|G|:G\in{\cal G}]$. When more convenient, the
exponential notation is used to describe the type of a GDD: a GDD of
type $g_1^{t_1}g_2^{t_2}\cdots g_s^{t_s}$ is a GDD where there are
exactly $t_i$ groups of size $g_i$, $i\in[s]$.

Suppose further ${\cal G}=\{G_1,G_2,\ldots G_s\}$ and ${\cal
H}=\{H_1,H_2,\ldots H_s\}$ is a collection of subsets of $X$ with
the property $H_i \subseteq G_i$, $0\le i\le s$. Let
$H=\bigcup_{i=1}^s H_i$. Then the quadruple $(X,{\cal G},{\cal
H},{\cal B})$ is an {\em incomplete group divisible design} (IGDD)
of index $\lambda$ when every $2$-subset of $X$ not contained in a
group or $H$ appears in exactly $\lambda$ blocks, and $|B\cap G|\leq
1$ and $|B\cap H|\le 1$ for all $B\in{\cal B}$ and $G\in{\cal G}$.
The {\em type} of an IGDD $(X,\{G_1,G_2,\ldots,G_s\},
\{H_1,H_2,\ldots,H_s\}, {\cal B})$ is the multiset
$[(|G_i|,|H_i|):1\le i\le s]$ and we use the exponential notation
when more convenient.

Let $k$, $g$, $u$, and $w$ be positive integers such that $k\mid g$
and $u\geq (k+1)w$.
 Let $R_i=\{r\in{\mathbb Z}: ig/k\leq r\leq (i+1)g/k-1\}$.
 An {\em incomplete holey} GBTD
 with block size $k$ and type $g^{(u,w)}$, denoted $\text{IHGBTD}\left(k,g^{(u,w)}\right)$,
 %is a $(\{k\},k-1)$-GDD $(X,\{ G_0,G_1,\ldots,G_{u-1}\},{\cal B})$ of type $g^u$,
 is a $(\{k\},k-1)$-IGDD $(X,\{ G_0,G_1,\ldots,G_{u-1}\}, \{\varnothing,\ldots, \varnothing, G_{u-w},$ $\ldots, G_{u-1}\},{\cal B})$
 of type $(g,0)^{u-w}(g,g)^w$,
 whose blocks are arranged
 in a $(gu/k)\times g(u-1)$ array $\sf A$,
 with rows and columns indexed by elements from the sets
 $\{0,1,\ldots, gu/k-1\}$ and $\{0,1,\ldots,g(u-1)-1\}$, respectively, such that
 the following properties are satisfied.
 \begin{enumerate}[(i)]
\item The $(gw/k)\times g(w-1)$ subarray whose rows are indexed by $r\in R_i$,
where $u-w\leq i\leq u-1$, and columns indexed by $c$, where
$g(u-w)\leq c\leq g(u-1)-1$, is empty.
\item For each $i$, $0\leq i\leq u-w-1$, the blocks in row $r\in R_i$ form a partial $k$-parallel class
over $X\setminus G_i$, and for each $i$, $u-w\leq i\leq u-1$, the
blocks in row $r\in R_i$ form a partial $k$-parallel class over
$X\setminus \left(\bigcup_{j=u-w}^{w-1} G_j\right)$.
\item For each $j$, $0\leq j\leq g(u-w)-1$, the blocks in column $j$ form a parallel class,
and for each $j$, $g(u-w)\leq j\leq g(u-1)-1$, the blocks in column
$j$ form a partial parallel class over $X\setminus
\left(\bigcup_{i=u-w}^{w-1} G_j\right)$.
 \end{enumerate}
Each group acts as a {\em hole} of the design, since no block
contains any $2$-subset of a group. The design is also {\em
incomplete} in the sense that the array ${\sf A}$ contains an empty
$(gw/k)\times g(w-1)$ subarray.

%\begin{example}
%Let ${\cal G}=\{G_0,G_1,\ldots,G_7\}$, where
%$G_0=\{\text{a},\text{b},\text{c}\}$, $G_1=\{\text{d},\text{e},\text{f}\}$,
%$G_2=\{\text{g},\text{h},\text{i}\}$, $G_3=\{\text{j},\text{k},\text{l}\}$,
%$G_4=\{\text{m},\text{n},\text{o}\}$, $G_5=\{\text{p},\text{q},\text{r}\}$,
%$G_6=\{1,2,3\}$, $G_7=\{4,5,6\}$, and $X=\bigcup_{i=0}^7 G_i$.
%Then
%\begin{center}
% {\footnotesize
% \setlength\tabcolsep{1pt}
% \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
%% \begin{tabular}{|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}}\hline
%\hline
% 1f j & 1dk & 1el & 6op & 6mq & 6nr & 5oe & 5mf & 5nd & 4jq & 4kr & 4lp & 3nh & 3oi & 3mg & 2hl & 2ij & 2gk & egr & fhp & diq \\ \hline
% 2ko & 2lm & 2jn & 1im & 1gn & 1ho & 6ra & 6pb & 6qc & 5rh & 5pi & 5qg & 4mb & 4nc & 4oa & 3qk & 3rl & 3pj & hjc & ika & glb \\ \hline
% 3bn & 3co & 3am & 2nr & 2op & 2mq & 1lp & 1jq & 1kr & 6cd & 6ae & 6bf & 5ck & 5al & 5bj & 4pe & 4qf & 4rd & kmf & lnd & joe \\ \hline
% 4ah & 4bi & 4cg & 3eq & 3fr & 3dp & 2qc & 2ra & 2pb & 1oa & 1mb & 1nc & 6fg & 6dh & 6ei & 5fn & 5do & 5em & npi & oqg & mrh \\ \hline
% 5iq & 5gr & 5hp & 4dk & 4el & 4fj & 3hb & 3ic & 3ga & 2bf & 2cd & 2ae & 1rd & 1pe & 1qf & 6ij & 6gk & 6hl & qal & rbj & pck \\ \hline
% 6lm & 6jn & 6ko & 5lb & 5jc & 5ka & 4gn & 4ho & 4im & 3ke & 3lf & 3jd & 2ei & 2fg & 2dh & 1cg & 1ah & 1bi & bdo & cem & afn \\ \hline
% ecr & fap & dbq & hfc & ida & geb & kif & lgd & jhe & nli & ojg & mkh & qol & rmj & pnk & bro & cpm & aqn & & & \\ \hline
% dgp & ehq & fir & gja & hkb & ilc & jmd & kne & lof & mpg & nqh & ori & paj & qbk & rcl & adm & ben & cfo & & & \\ \hline
% \end{tabular}
% }
% \end{center}
%is an $\text{IHGBTD}(3,3^{(8,2)})$.
%\end{example}

\noindent We note that an $\text{IHGBTD}(k,g^{(u,1)})$ is a holey
GBTD, $\text{HGBTD}(k,g^u)$, as defined by Yin et al. \cite{YYW}.
The following was established by Yin et al. \cite{YYW}.


% When $w=1$, an IHGBTD$(k, g^{(u,w)})$ is said to be a {\em holey GBTD} or {\em HGBTD$(k,g^{u})$},
% which was first introduced in \cite{YYW} and proved to be very useful to construct GBTDs.

 \begin{prop}[Yin et al. \cite{YYW}]\label{HGBTD+GBTD coro}
  If there exists an $\text{\rm HGBTD}(k, k^u)$, then there exists a $\text{\rm GBTD}(k, u)$.
 \end{prop}

Proposition \ref{HGBTD+GBTD coro} shows that a $\text{GBTD}(k,u)$
can be obtained from an $\text{HGBTD}(k,k^u)$. The result below
shows how we can obtain an $\text{HGBTD}(k,g^u)$ (and, in
particular, an $\text{HGBTD}(k,k^u)$ from an
$\text{IHGBTD}(k,g^{(u,w)})$ and an $\text{HGBTD}(k,g^w)$.

 \begin{prop}\label{IHGBTD fill in hole}
 If there exist an $\text{\rm IHGBTD}(k, g^{(u,w)})$
 and an $\text{\rm HGBTD}(k,g^w)$, then there exists an $\text{\rm HGBTD}(k,g^u)$.
 \end{prop}

 \noindent {\bf Proof:}
 When $w=1$, an $\text{HGBTD}(k,g^w)$ is empty and an
 $\text{IHGBTD}(k,g^{(u,w)})$ is just an HGBTD$(k,g^u)$. So assume $w>1$ and
 let $(X,{\cal G},{\cal B})$ be an $\text{IHGBTD}(k, g^{(u,w)})$
  with ${\cal G}=\{G_0,G_1,\ldots, G_{u-1}\}$. Fill in the empty subarray of this IHGBTD
  with an $\text{HGBTD}(k,g^w)$, $(X',{\cal G}',{\cal B}')$, with
  ${\cal G}'=\{G_{u-w},G_{u-w+1},\ldots,G_{u-1}\}$ and $X'=\bigcup_{i=u-w}^{u-1} G_i$.
  The resulting array is a $\text{HGBTD}(k,g^u)$, $(X,{\cal G},{\cal B}\cup{\cal B}')$.
\hfill $\Box$

 \section{Starter-Adder Construction for IHGBTD}

 The starter-adder technique first used by Mullin and Nemeth {\cite{MullinNemeth:1969}}
 to construct Room squares (also a combinatorial array) has been
 useful in constructing many types of designs with
 orthogonality properties, including GBTDs (see \cite{Lamken1,RV,YY1,YY2,YYW}).
 Here, we extend the technique to the construction of
 IHGBTDs. Since only IHGBTD$(k,g^{(u,w)})$ with $g=k$ are considered here, we
 describe our construction for this case.

 Let $\Gamma$ be an additive abelian group of
 order $k(u-w)$ with $u\geq (k+1)w$, and let
 $\Gamma_0\subseteq \Gamma$ be a subgroup of order $k$.
% Fix a set, $\{h_0 = 0, h_1,\ldots, h_{u-w-1}\}\subseteq\Gamma$, of representatives for the cosets
% of $\Gamma_0$ so that $\Gamma_i = \Gamma_0+h_i$, $0\leq i\leq u-w-1$,
% are the cosets of $\Gamma_0$.
 Fix a set, $\Delta=\{\delta_0 = 0, \delta_1,\ldots, \delta_{u-w-1}\}\subseteq\Gamma$, of representatives for the cosets
 of $\Gamma_0$ so that $\Gamma_i = \Gamma_0+\delta_i$, $0\leq i\leq u-w-1$,
 are the cosets of $\Gamma_0$.
 Let $H$ be a set of $kw$ points such that $H$ and $\Gamma$ are disjoint.
 Further, let $H$ be partitioned into $w$ subsets $H_0,H_1,\ldots,H_{w-1}$ of size
 $k$ each.

 We take $X=\Gamma\bigcup H$ to be the point set of an
 $\text{IHGBTD}(k,k^{(u,w)})$.
 An {\em intransitive starter} for an $\text{IHGBTD}(k,k^{(u,w)})$,
with groups $\{G_0,G_1,\ldots,G_{u-1}\}$, where % $(X,\{G_0,G_1,\ldots,G_{u-1}\},{\cal B})$, where
 \begin{equation*}
 G_i=\begin{cases}
 \Gamma_i,&\text{if $0\leq i\leq u-w-1$;} \\
 H_{i-u+w},&\text{if $u-w\leq i\leq u-1$,}
 \end{cases}
 \end{equation*}
 is defined as a quadruple $(X,{\cal S},{\cal R},{\cal C})$ satisfying the
 properties:

 \begin{enumerate}[(i)]
 \item $(X,{\cal S})$, $(X,{\cal R})$, and $(X,{\cal C})$ are $\{k\}$-uniform set systems of size
 $u-w$, $w$, and $w-1$, respectively;
 \item among the blocks in $\cal S$, $kw$ of them intersects $H$ in one point, that is,
 $|\{B\in{\cal S}: |B\cap H|=1\}|=kw$;
 %\item $S$ consists of exactly $u-w$ base blocks of size $k$, denoted by $B_0,B_1,\cdots, B_{(u-w-1)}$,
% in which there are $kw$ base blocks each containing one infinite point in
% $H$;
\item blocks in $\cal R$ are each disjoint from $H$;
% \item $R$ consists of $w$ base blocks over $G$, denoted by $R_1,R_2\cdots, R_w$,
% in which each contains no infinite points;
\item blocks in $\cal C$ are each disjoint from $H$, and
$\bigcup_{i=0}^{u-w-1} (\delta_i+C)=\Gamma$, for each $C\in{\cal
C}$.
% \item $C$ consists of $w-1$ base blocks over
% $G$, denoted by $C_0,C_1\cdots, C_{w-2}$, in which each contains no infinite
% points, satisfying $\bigcup_{j=0}^{u-w-1} (C_i +h_j)=G$ for each $0\leq
% i<w-1$.
 \item ${\cal S}\bigcup{\cal R}$ is a partition of $X$;
% \item the difference list from the base blocks of $\mathcal S\bigcup \mathcal R\bigcup \mathcal C$ contains every element of $G\backslash G_0$ precisely $k
% -1$ times, and no element in $G_0$.
 \item the difference list from the base blocks of $\mathcal S\bigcup \mathcal R\bigcup \mathcal C$ contains
  every element of $\Gamma\setminus \Gamma_0$ precisely $k -1$ times, and no element in $\Gamma_0$.
 \end{enumerate}

Suppose $\mathcal S=\{B_0,B_1,\ldots, B_{u-w-1}\}$. Then a
corresponding {\em adder} $\Omega(\mathcal S)$ for $\mathcal S$ is a
permutation
 $\Omega(\mathcal S) = \left(\omega_0, \omega_1,\ldots, \omega_{u-w-1}\right)$ of the $u-w$ elements of the representative system
 %$\{\gamma_0, \gamma_1,\ldots, \gamma_{u-w-1}\}$,
 $\Delta$ satisfying the following property:
 \begin{enumerate}[(vii)]
 \item the multiset $\left(\bigcup_{i=0}^{u-w-1} (B_i +\omega_i)\right)\bigcup \left(\bigcup_{C\in \mathcal C} C\right)$
 contains exactly $k$ elements (not necessarily distinct) from  $\Gamma_j$ for $1\leq j \le u-w-1$,
 and is disjoint from $\Gamma_0$. We remark that when $B\in \mathcal S$ and $B\cap H=\{\infty\}$, or $B=\{\infty,b_1,b_2,\ldots,b_{k-1}\}$,
the block $B+\gamma$ is defined to be
$\{\infty,b_1+\gamma,b_2+\gamma,\ldots,b_{k-1}+\gamma\}$ for
$\gamma\in\Gamma$.
\end{enumerate}
%
%
% A corresponding {\em adder} $A(\cal S)$ for $\cal S$ is a permutation
% $A(\mathcal S) = (a_0, a_1,\cdots, a_{(u-w-1)})$ of the $u-w$ elements of the representative system
% $\{\gamma_0, \gamma_1,\cdots, \gamma_{u-w-1}\}$.
% The union of the blocks
% $(\cup_{i=0}^{u-w-1} (B_i +a_i))\bigcup (\cup_{i=0}^{w-2}C_i)$ contains exactly $k$ elements (not necessarily
% distinct) from any group $G_j$ for $1\leq j< u-w$, and no element of $G_0$. The
% addition $B_i+a_i$ is performed in $G$ with the infinite point of
% $H$ fixed whenever it occurs in $B_i$ for each $0\leq i< u-w$.

The result below shows how to construct an IHGBTD from an
intransitive starter and its corresponding adder.

 \begin{prop}\label{starter-adder-IHGBTD}
 Let $\Gamma$ be an additive abelian group of order $k(u-w)$ with $u \ge (k + 1)w$
 and $\Gamma_0$ be a subgroup of order $k$.
 Define $X$ and the groups $G_i$ $(0\leq i\leq u-1)$ as above.
 %, $\Delta$, $H$, $G_i$ $(0\leq i\leq u-1)$ as above.
 If there exists an intransitive starter $(X, \mathcal S,\mathcal R, \mathcal C)$ with groups $\{G_i: 0\le i\le u-1\}$,
 %groups $G_i$ $(0\leq i\leq u-1)$ defined above
  a corresponding
 adder $\Omega(\cal S)$, then there exists an IHGBTD$(k,k^{(u,w)})$.
 \end{prop}
\noindent {\bf Proof:}
 Retain the notations in the definition of intransitive starter and adder.
 Suppose
\begin{equation*}
 \mathcal A=\left\{ A+\gamma: \gamma\in \Gamma, A\in \mathcal S\cup\mathcal R\cup\mathcal C\right\},
\end{equation*}
\noindent then $(X,\{G_0,G_1,\ldots,G_{u-1}\},\{\varnothing,\ldots,
\varnothing, H_0,\ldots,H_{w-1}\},\mathcal A)$ forms a
$(\{k\},k-1)$-IGDD of type $(k,0)^{u-w}(k,k)^w$ by Condition (vi) in
the definition of intransitive starter. Therefore, it remains to
arrange the blocks in a $u\times k(u-1)$ array.

  First, consider the blocks $\cal S$. Consider a $(u-w)\times (u-w)$ array $\sf S$,
  whose rows and columns are indexed with the elements in $\Delta$.
  Now identify the elements in $\Delta$ with elements in the quotient group $\Gamma/\Gamma_0$,
  so that the binary operation $\mathring +$ on $\Delta$ is defined by the additive operation on $\Gamma/\Gamma_0$.
  In addition, for $\delta\in \Delta$, denote the additive inverse of $\delta$ by $\mathring - \delta$.
  That is, $\delta\mathring + (\mathring - \delta)=\delta_0$.

  So, for $0\le i,j\le u-w-1$, we place the block $B_i+\delta_j$ at the cell $(\delta_j \mathring - \delta_l,\delta_j)$ if $\delta_l=\omega_i$.
  Note that this placement is well defined because $\Omega(\mathcal S)$ is a permutation of $\Delta$.
  Let $\Gamma_0=\{\gamma_0=0,\gamma_1,\cdots,\gamma_{k-1}\}$. Form a $(u-w)\times
 k(u-w)$ array $\widehat{\sf S}$ from the square $\sf S$ by concatenating $k$ squares ${\sf D}+\gamma_i$
 where $0\leq i\le k-1$ as follows.
  \begin{equation*}
  \widehat{\sf S}=
  \begin{array}{|cccc|}
  \hline
  {\sf S} & {\sf S}+\gamma_1&\cdots& {\sf S}+\gamma_{k-1}\\
  \hline
  \end{array}
  \end{equation*}

 Next, let ${\cal R}=\{R_1,R_2,\ldots, R_w\}$ and construct a $w\times k(u-w)$
  array $\widehat{\sf R}$ in the following way:
  \begin{equation*}
   \widehat{\sf R} =
  \begin{array}{|cccc|}
  \hline
  {\sf R}&{\sf R}+\gamma_1&\cdots&{\sf R}+\gamma_{k-1}\\
  \hline
  \end{array}\ ,
  \end{equation*}
  \noindent where the $w\times w$ subarray $\sf R$ is given by
  \begin{equation*}
  {\sf R}  =
  \begin{array}{|cccc|}
  \hline
  R_1&R_1+\delta_1&\cdots&R_1+\delta_{u-w-1}\\
  R_2&R_2+\delta_1&\cdots&R_2+\delta_{u-w-1}\\
 \vdots &\vdots &\ddots &\vdots \\
  R_w&R_w+\delta_1&\cdots&R_w+\delta_{u-w-1}\\
  \hline
  \end{array}\ .
  \end{equation*}

   Similarly, let ${\cal C}=\{C_0,C_1,\ldots, C_{w-2}\}$, and we construct a $(u-w)\times k(w-1)$
  array $\widehat{\sf C}$.
  \begin{equation*}
  \widehat{\sf C}=
  \begin{array}{|cccc|}
  \hline
 {{\sf C}_0}&{{\sf C}_1}& \cdots &{{\sf C}_{w-2}}\\
  \hline
  \end{array}\ ,
  \end{equation*}
  \noindent where each $(u-w)\times k$ subarray ${\sf C}_i$, $0\le i\le w-2$, is given by
 \begin{equation*}
  {\sf C}_i=
  \begin{array}{|cccc|}
  \hline
  C_i&C_i+\gamma_1&\cdots&C_i+\gamma_{k-1}\\
  C_i+\delta_1&C_i+\delta_1+\gamma_1&\cdots&C_i+\delta_1+\gamma_{k-1}\\
 \vdots &\vdots &\ddots &\vdots \\
  C_i+\delta_{u-w-1}&C_i+\delta_{u-w-1}+\gamma_1&\cdots&C_i+\delta_{u-w-1}+\gamma_{k-1}\\
  \hline
  \end{array}\ .
  \end{equation*}

  Finally, let
  \begin{equation*}
  \renewcommand\arraystretch{1.5}
  {\sf A}=
  \begin{array}{|ccc|ccc|}
  \hline
  &\widehat{\sf S}&& &\widehat{\sf C}&\\\hline
  &\widehat{\sf R}&& &&\\
  \hline
  \end{array}\ ,
  \end{equation*}
    \noindent and it is readily verified that the placement results in an IHGBTD$(k,k^{(u,w)})$. \qed
  % In addition, let
% \begin{align*}
% \mathcal S &=\{B_0,B_1,\ldots, B_{u-w-1}\},\\
% \mathcal R &=\{R_1,R_2,\ldots, R_{w}\},\\
% \mathcal C &=\{C_0,C_1,\ldots, C_{w-2}\},\\
% \end{align*}
%
%
%
% First, we use the starter $S$ and adder $A(S)$ to construct a square $K$ of
% side $u-w$ whose rows and columns are indexed with the elements $h_0,h_1,\cdots, h_{u-w-1}$.
% For any $h_r\in\{h_0,h_1,\cdots,h_{u-w-1}\}$, place the block $B_i$
% in the cell $(-h_r,0)$ if and only if the corresponding adder $a_i$
% of this base block is $h_r$. Here we identify $h_r$ with a certain $h_j$ $(0\leq j < u-w)$ whenever $h_r \in h_j + G_0$.
% This is always able to be done, as $A(S)$ is a permutation of the representatives
% $h_0, h_1, \cdots h_{u-w-1}$ and $\{h_0 = 0, -h_1, -h_2,\cdots, -h_{u-w-1}\}$ is
% obviously another system of representatives of the cosets of $G_0$ in $G$.
% For the remaining columns $h_c \in \{h_1, h_2,\cdots, h_{u-w-1}\}$, if $B$ is the block in the cell $(h_r-h_c, 0)$,
% we place $B + h_c$ to the cell $(h_r, h_c)$.
% Here $h_r-h_c = h_j$ if and only if $h_r-h_c \in h_j + G_0$ $(0\leq  j< u-w)$.
%
% Suppose that $G_0=\{g_0=0,g_1,\cdots,g_{k-1}\}$. Form a $(u-w)\times
% k(u-w)$ array $\widehat{K}$ from the square $K$ by concatenating $k$ squares $K+g_i$
% where $0\leq i<k$ as follows.
%  $$
%  \widehat{K}=
%  \begin{array}{|c|c|c|c|ccc}
%  \hline
%  K&K+g_1&\cdots&K+g_{k-1}\\
%  \hline
%  \end{array}
%  $$

%  Next, utilize $R_i$ $(1\leq i\leq w)$ to construct a $w\times k(u-w)$
%  array $\widehat{R}$ in the following way.
%   $$
%  \widehat{R}=
%  \begin{array}{|c|c|c|c|ccc}
%  \hline
%  R&R+g_1&\cdots&R+g_{k-1}\\
%  \hline
%  \end{array}
%  $$
%  where
%  $$
%  R=
%  \begin{array}{|c|c|c|c|ccc}
%  \hline
%  R_1&R_1+h_1&\cdots&R_1+h_{u-w-1}\\\hline
%  R_2&R_2+h_1&\cdots&R_2+h_{u-w-1}\\\hline
% \vdots &\vdots &\ddots &\vdots \\\hline
%  R_w&R_w+h_1&\cdots&R_w+h_{u-w-1}\\
%  \hline
%  \end{array}
%  $$

%  Then, we use $C_i$ $(0\leq i< w-1)$ to construct a $(u-w)\times k(w-1)$
%  array $\widehat{C}$ in the following way.
%  $$
%  \widehat{C}=
%  \begin{array}{|c|c|c|c|ccc}
%  \hline
%  \widehat{C_0}& \widehat{C_1}& \cdots &\widehat{C_{w-2}}\\
%  \hline
%  \end{array}
%  $$
%  where
%  $$
%  \widehat{C_i}=
%  \begin{array}{|c|c|c|c|ccc}
%  \hline
%  C_i&C_i+g_1&\cdots&C_i+g_{k-1}\\\hline
%  C_i+h_1&C_i+h_1+g_1&\cdots&C_i+h_1+g_{k-1}\\\hline
% \vdots &\vdots &\ddots &\vdots \\\hline
%  C_i+h_{u-w-1}&C_i+h_{u-w-1}+g_1&\cdots&C_i+h_{u-w-1}+g_{k-1}\\
%  \hline
%  \end{array}
%  $$
%  for $0\leq i<w-1$.


%  Finally, let $$
%  \widehat{M}=
%  \begin{array}{|ccc|ccc|}
%  \hline
%  &\widehat{K}&& &\widehat{C}&\\\hline
%  &\widehat{R}&& &&\\
%  \hline
%  \end{array}
%  $$
%  Note that all the infinite points in $H$ are
%  always fixed whenever we do arithmetic.
%  Then it is readily checked the $u\times k(u-1)$ array $\widehat{M}$
%  forms an IHGDTB$(k,k^{(u,w)})$, which contains a $w\times k(w-1)$
%  empty array in the lower right corner.




 \section{Proof of Theorem 1.2}

We first remove all the eight remaining values in Theorem \ref{old
GBTD}.
% In this section, we utilize Proposition \ref{starter-adder-IHGBTD} to remove all the eight remaining values in Theorem
% \ref{old GBTD} and then prove our main result.

 \begin{lemma}\label{IHGBTD class1}
  For $(u,w)\in\{(28,5),(32,5),$ $(33,6)\}$, an IHGBTD$\left(4,4^{(u,w)}\right)$ exists.
 \end{lemma}
 \noindent {\bf Proof:}
 We apply Proposition \ref{starter-adder-IHGBTD} to construct the desired IHGBTDs.
 Take
 \begin{align*}
 \Gamma & = \mathbb{Z}_{u-w}\times \mathbb{Z}_4,\\
 \Gamma_0 & = \{0\}\times \mathbb{Z}_4,\\
 \Delta &= \{(0, 0), (1, 0), \ldots, (u-w-1, 0)\}, \mbox{ and }\\
 H &= \bigcup_{i=0}^{w-1}H_{i}, \mbox{ where $H_i=\{\infty_{i},\infty_{i+w},\infty_{i+2w},\infty_{i+3w}\}$ for $0\leq i\le w-1$}.
 \end{align*}
% $\Gamma = \mathbb{Z}_{u-w}\times \mathbb{Z}_4$
% and $H=\bigcup_{i=0}^{w-1}H_{i}$ where $H_i=\{\infty_{i},\infty_{i+w},\infty_{i+2w},\infty_{i+3w}\}$ for $0\leq i\le w-1$.
% Then $\Gamma_0 = \{0\}\times \mathbb{Z}_4$ is an additive
% subgroup of order four in $\Gamma$. Take the representative system of cosets $\Delta$ to be $\{(0, 0), (1, 0), \cdots, (u-w-1, 0)\}$.
 For each pair $(u,w)\in\{(28,5),(32,5),$ $(33,6)\}$, the desired intransitive starter and corresponding adder are displayed below.
 Here we write the element $(a,b)$ of $\Gamma$ as $a_b$ for succinctness.

 When $(u,w)=(28,5)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
  \{4_1,  3_0,  7_0,  0_0\}    &17_{0}   &\{5_0, 19_0, 12_1,  1_2\}  &12_{0}  &\{18_0, 13_3, 16_3,  8_1\}    &19_{0}\\
  \{\infty_{0},  3_1, 12_2, 11_3\}    & 1_{0}   &\{\infty_{1},14_3,  6_0, 10_3\}  &21_{0}   &\{\infty_{2}, 14_1,  9_1, 20_1\}    &20_{0}\\
  \{\infty_{3}, 19_1, 10_1, 22_2\}    & 7_{0}   &\{\infty_{4}, 3_3,  1_3,  2_2\}  &18_{0}   &\{\infty_{5},  0_2, 15_1,  1_0\}    &15_{0}\\
  \{\infty_{6},  1_1,  6_3,  9_3\}    & 2_{0}   &\{\infty_{7},14_0, 11_1,  0_1\}  &10_{0}   &\{\infty_{8},  0_3, 17_2, 21_2\}    &22_{0}\\
  \{\infty_{9},  4_3,  8_0, 21_0\}    & 6_{0}   &\{\infty_{10},13_1, 19_3, 16_2\}  & 9_{0}  &\{\infty_{11},  4_2, 21_3, 17_1\}   & 5_{0}\\
  \{\infty_{12}, 17_0,  5_2, 21_1\}    &16_{0}  &\{\infty_{13}, 5_1, 20_2, 11_2\}  & 4_{0}  &\{\infty_{14}, 22_0,  2_3, 16_0\}   &14_{0}\\
  \{\infty_{15}, 18_3, 20_3,  2_0\}    & 0_{0}  &\{\infty_{16},12_3,  2_1, 22_3\}  & 3_{0}  &\{\infty_{17},  5_3,  7_1, 17_3\}   & 8_{0}\\
  \{\infty_{18},  6_2,  9_0, 19_2\}    &13_{0}  &\{\infty_{19}, 7_2,  8_3, 22_1\}  &11_{0}  &&\\ \hline
 \end{array}
 \end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
{\cal C}=&\{18_0, 11_1,  5_3,  6_2\}, &\{18_2,  8_3, 19_0,  6_1\},&\{14_3, 12_0,  3_2,  7_1\},\\
&\{5_2,  7_1, 16_3, 11_0\}.\\
{\cal R}=&\{3_2, 18_2, 16_1, 10_2\},&\{8_2, 15_0, 20_0, 13_2\},&\{13_0,  9_2, 18_1, 15_3\},\\
&\{6_1,  7_3, 14_2, 15_2\},&\{12_0, 10_0,  4_0, 11_0\}.
\end{array}
  \end{equation*}
 }
 \vskip 5pt
When $(u,w)=(32,5)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
 %\begin{array}{ccccccc}
 %&&&(u,w)=(32,5)&&&\\
% \hline
%      &S &A(S) &S &A(S) &S &A(S)\\
% \hline
% &\{13_1,  7_3, 11_2, 23_1\} &19_{0}&\{24_0, 19_1,  8_3, 10_2\} &26_{0}&\{8_1, 22_3,  5_3, 14_0\}   &23_{0}\\
 \{ 4_2, 17_2, 16_1, 22_2\} &16_{0}&\{ 3_1,  4_1,  1_0,  9_1\} &11_{0}&\{4_3, 26_3, 22_0, 10_3\}   & 0_{0}\\
 \{14_1,  6_0, 26_0,  3_0\} &12_{0}&\{\infty_{0},  3_3, 24_2, 25_1\} & 7_{0}&\{\infty_{1},  2_2, 12_0,  1_3\}   & 6_{0}\\
 \{\infty_{2},  0_1, 26_1, 20_2\} & 4_{0}&\{\infty_{3}, 25_0, 15_0, 23_0\} &15_{0}&\{\infty_{4}, 13_0, 21_2, 16_0\}   & 3_{0}\\
 \{\infty_{5},  5_0, 19_3, 12_1\} &24_{0}&\{\infty_{6},  6_3, 14_3, 13_2\} & 1_{0}&\{\infty_{7},  1_2,  2_0,  0_0\}   &21_{0}\\
 \{\infty_{8},  0_2, 10_0, 19_0\} &14_{0}&\{\infty_{9}, 15_2, 18_2,  0_3\} & 2_{0}&\{\infty_{10},  6_1,  5_2,  2_3\}   &17_{0}\\
 \{\infty_{11}, 12_3, 25_2, 11_3\} &22_{0}&\{\infty_{12}, 10_1, 21_3, 17_3\} &18_{0}&\{\infty_{13}, 17_0,  9_0, 20_3\}   &20_{0}\\
 \{\infty_{14}, 20_0,  3_2, 16_3\} & 5_{0}&\{\infty_{15}, 12_2, 21_1,  8_2\} & 9_{0}&\{\infty_{16}, 18_1, 11_0, 15_3\}   &10_{0}\\
 \{\infty_{17},  1_1, 15_1, 17_1\} & 8_{0}&\{\infty_{18},  9_2, 16_2, 23_2\} &13_{0}&\{\infty_{19}, 14_2, 18_3, 21_0\}   &25_{0}\\
 \hline
  \end{array}
 \end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
 {\cal C}=& \{ 1_3, 26_0, 16_1, 17_2\},&\{ 5_3, 14_1, 24_2, 12_0\},&\{19_2, 25_0, 17_1, 13_3\},\\
   &\{ 6_2,  8_0, 11_3, 13_1\}.\\
 {\cal R}=&\{ 5_1, 11_1, 24_3, 20_1\}, &\{24_1, 18_0,  7_0,  6_2\},&\{22_1, 25_3,  8_0, 13_3\},\\
   &\{19_2,  7_2,  2_1, 23_3\},&\{ 7_1,  9_3, 26_2,  4_0\}.
 \end{array}
 \end{equation*}
 }
 \vskip 5pt

When $(u,w)=(33,6)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
% $$
% {\footnotesize
% \begin{array}{ccccccc}
% &&&(u,w)=(33,6)&&&\\
% \hline
%      &S &A(S) &S &A(S) &S &A(S)\\
% \hline
 \{22_0,  0_1, 23_0, 21_3\}    &13_{0}&\{25_3,  4_3, 15_1, 20_1\}    & 4_{0}&\{7_3,  2_2, 23_3,  1_0\}   & 7_{0}    \\
 \{\infty_{0},  21_1,  3_0, 22_2\}    &18_{0}&\{\infty_{1},  0_0, 14_3, 10_1\}    & 6_{0}&\{\infty_{2},  12_3,  8_0, 16_1\}   & 8_{0}    \\
 \{\infty_{3},   6_1, 23_2,  9_1\}    &23_{0}&\{\infty_{4},  4_0,  8_2, 14_2\}    & 2_{0}&\{\infty_{5},  14_1,  2_3,  6_0\}   &17_{0}    \\
 \{\infty_{6},  21_2, 24_2, 11_2\}    & 9_{0}&\{\infty_{7},  5_0,  2_1, 25_1\}    &20_{0}&\{\infty_{8},  11_1, 22_1, 12_1\}   &22_{0}    \\
 \{\infty_{9},   0_2,  7_2, 19_2\}    &15_{0}&\{\infty_{10}, 13_0, 16_0, 14_0\}    &24_{0}&\{\infty_{11}, 11_0, 15_0, 18_1\}   & 3_{0}    \\
 \{\infty_{12},  7_0,  9_0, 26_1\}    &19_{0}&\{\infty_{13}, 25_0,  7_1, 10_0\}    &21_{0}&\{\infty_{14}, 18_0, 25_2, 26_3\}   &26_{0}    \\
 \{\infty_{15},  4_2, 15_2, 13_3\}    &16_{0}&\{\infty_{16}, 17_1, 20_0, 11_3\}    & 5_{0}&\{\infty_{17}, 20_2,  9_3, 12_0\}   &14_{0}    \\
 \{\infty_{18}, 26_2,  5_2, 17_2\}    &12_{0}&\{\infty_{19}, 24_0, 13_1, 10_3\}    & 1_{0}&\{\infty_{20},  1_3, 10_2, 12_2\}   &11_{0}    \\
 \{\infty_{21},  3_2, 15_3, 24_1\}    &25_{0}&\{\infty_{22},  5_1, 18_3, 21_0\}    &10_{0}&\{\infty_{23}, 17_0, 24_3, 26_0\}   & 0_{0}    \\
  \hline
  \end{array}
 \end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
 {\cal C}= &\{ 3_3, 10_1,  5_2, 15_0\},&\{ 8_3, 14_1,  9_2, 18_0\},&\{12_0, 10_3, 26_2,  5_1\},\\
    &\{21_2, 11_1, 23_0, 9_3\},&\{15_1,  5_2, 12_3,  3_0\}.\\

 {\cal R}= &\{ 6_3,  2_0, 18_2, 19_0\},&\{ 8_3,  9_2,  3_1,  1_2\} ,&\{17_3,  3_3,  4_1, 22_3\},\\
    &\{19_3, 13_2,  6_2,  5_3\},&\{16_3, 23_1,  1_1, 19_1\} ,&\{20_3, 16_2,  8_1,  0_3\}.\\
 \end{array}
 \end{equation*}
 } \qed

 \begin{lemma}\label{IHGBTD class2}
  For $(u,w)\in \{(34,6),(44,8)\}$, an IHGBTD$\left(4,4^{(u,w)}\right)$ exists.
 \end{lemma}
 \noindent {\bf Proof:}
 As with Lemma \ref{IHGBTD class1}, we apply Proposition \ref{starter-adder-IHGBTD} to construct the desired IHGBTDs.
  Take
 \begin{align*}
 \Gamma & = \mathbb{Z}_{2(u-w)}\times \mathbb{Z}_2,\\
 \Gamma_0 & = \{0, u-w\}\times \mathbb{Z}_2,\\
 \Delta &= \{(0, 0), (1, 0), \cdots, (u-w-1, 0)\}, \mbox{ and }\\
 H &= \bigcup_{i=0}^{w-1}H_{i}, \mbox{ where $H_i=\{\infty_{i},\infty_{i+w},\infty_{i+2w},\infty_{i+3w}\}$ for $0\leq i\le w-1$}.
 \end{align*}
% here take $\Gamma = \mathbb{Z}_{2(u-w)}\times \mathbb{Z}_2$
% and $H=\bigcup_{i=0}^{w-1}H_{i}$ where $H_i=\{\infty_{i},\infty_{i+w},\infty_{i+2w},\infty_{i+3w}\}$ for $0\leq i\le w-1$.
% Then $\Gamma_0 = \{0, u-w\}\times \mathbb{Z}_2$ is an subgroup of order four in $\Gamma$.
% %Take the representative system of the cosets of $G_0$
% Choose $\Delta$ to be $\{(0, 0), (1, 0), \cdots, (u-w-1, 0)\}$.
 The desired intransitive starter and corresponding adder for $(u,w)\in \{(34,6),(44,8)\}$ are displayed below.
 Here we write the element $(a,b)$ of $\Gamma$ as $a_b$ for succinctness.

When $(u,w)=(34,6)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
% $$
% {\footnotesize
% \begin{array}{ccccccc}
% &&&(u,w)=(34,6)&&&\\
% \hline
%      &S &A(S) &S &A(S) &S &A(S)\\
% \hline
 \{41_1, 16_0,  6_0, 15_0\}       &20_{0}&\{36_0,  9_0, 33_1, 13_1\}       &16_{0}&\{37_0, 18_0, 26_1,  4_1\}       & 0_{0}\\
 \{16_1,  2_1,  4_0,  3_1\}       & 3_{0}&\{\infty_{0}, 20_1, 24_0, 42_0\} &23_{0}&\{\infty_{1}, 22_1, 30_0, 39_1\} &11_{0}\\
 \{\infty_{2}, 14_0, 31_1,  1_1\} &10_{0}&\{\infty_{3}, 48_0, 45_0,  8_0\} &25_{0}&\{\infty_{4}, 25_1, 48_1, 14_1\} & 4_{0}\\
 \{\infty_{5},  8_1, 30_1, 20_0\} &12_{0}&\{\infty_{6},  6_1, 21_0, 44_1\} & 2_{0}&\{\infty_{7}, 40_1, 33_0, 52_1\} & 1_{0}\\
 \{\infty_{8}, 45_1, 21_1, 28_1\} &18_{0}&\{\infty_{9}, 27_0, 28_0, 34_1\} &17_{0}&\{\infty_{10}, 42_1, 35_1, 37_1\}&22_{0}\\
 \{\infty_{11},  3_0, 22_0, 12_0\}&19_{0}&\{\infty_{12}, 44_0, 35_0, 39_0\}&14_{0}&\{\infty_{13}, 36_1,  7_0,  9_1\}& 7_{0}\\
 \{\infty_{14}, 15_1, 53_1, 51_1\}& 6_{0}&\{\infty_{15}, 53_0, 11_0, 51_0\}&15_{0}&\{\infty_{16}, 50_0, 55_1, 10_1\}& 9_{0}\\
 \{\infty_{17}, 52_0, 32_1, 17_1\}&13_{0}&\{\infty_{18}, 55_0, 29_1, 25_0\}& 5_{0}&\{\infty_{19},  0_1,  7_1, 41_0\}&27_{0}\\
 \{\infty_{20}, 12_1, 31_0, 47_0\}& 8_{0}&\{\infty_{21}, 17_0, 27_1, 47_1\}&21_{0}&\{\infty_{22}, 19_0, 23_0, 29_0\}&24_{0}\\
 \{\infty_{23}, 34_0, 40_0, 50_1\}&26_{0}&&&& \\
  \hline
  \end{array}
 \end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
 {\cal C}=&\{27_1, 10_0, 44_1, 51_0\},&\{35_1, 15_0, 50_0, 14_1\},&\{16_1, 51_1, 54_0, 27_0\},\\
    &\{24_1, 12_0, 37_0, 21_1\},&\{39_0,  2_1, 45_1, 50_0\}.\\

 {\cal R}= &\{13_0, 26_0, 38_0, 24_1\},&\{54_1, 23_1, 46_1, 49_1\},&\{ 1_0, 49_0, 18_1, 43_0\},\\
    &\{10_0,  2_0, 11_1, 54_0\},&\{46_0, 19_1, 43_1,  5_0\},&\{38_1, 32_0,  5_1,  0_0\}.\\
  \end{array}
 \end{equation*}
 }

\vskip 5pt When $(u,w)=(44,8)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
% {\footnotesize
% \begin{array}{ccccccc}
% &&&(u,w)=(44,8)&&&\\
% \hline
%      &S &A(S) &S &A(S) &S &A(S)\\
% \hline
  \{32_0, 69_1, 36_1, 53_1\}           &20_{0}   &\{42_1, 65_1,  0_0, 43_1\}         & 1_{0}  &\{39_1, 27_1, 45_1, 51_1\}           & 3_{0}\\
  \{22_1, 39_0, 55_1, 33_1\}           &11_{0}   &\{\infty_{0},  67_0, 40_1, 54_0\}  &22_{0}  &\{\infty_{1},  23_0, 10_1, 34_1\}    &25_{0}\\
  \{\infty_{2},  18_0, 67_1, 36_0\}    &28_{0}   &\{\infty_{3},  25_1, 10_0, 28_1\}  &16_{0}  &\{\infty_{4},  63_1,  6_0, 37_0\}    &29_{0}\\
  \{\infty_{5},  16_0, 44_0,  2_0\}    &35_{0}   &\{\infty_{6},  28_0, 50_1, 35_1\}  &10_{0}  &\{\infty_{7},  43_0, 46_1, 32_1\}    & 9_{0}\\
  \{\infty_{8},  69_0, 52_1,  2_1\}    &13_{0}   &\{\infty_{9},  37_1, 66_0, 71_1\}  &26_{0}  &\{\infty_{10}, 70_1, 21_1, 24_1\}    & 8_{0}\\
  \{\infty_{11}, 71_0, 15_1, 47_0\}    &32_{0}   &\{\infty_{12}, 59_0, 19_1,  6_1\}  &23_{0}  &\{\infty_{13},  9_0, 47_1, 20_0\}    & 7_{0}\\
  \{\infty_{14}, 52_0, 46_0, 60_1\}    &24_{0}   &\{\infty_{15}, 17_0, 60_0, 22_0\}  & 0_{0}  &\{\infty_{16}, 64_0, 54_1, 12_0\}    &17_{0}\\
  \{\infty_{17}, 49_0,  9_1, 53_0\}    & 4_{0}   &\{\infty_{18}, 68_0,  0_1, 56_1\}  &15_{0}  &\{\infty_{19}, 27_0, 12_1,  4_1\}    &27_{0}\\
  \{\infty_{20}, 65_0, 68_1, 23_1\}    & 2_{0}   &\{\infty_{21}, 20_1, 18_1,  8_0\}  &31_{0}  &\{\infty_{22}, 59_1, 17_1, 44_1\}    &14_{0}\\
  \{\infty_{23},  1_0, 70_0, 26_1\}    &12_{0}   &\{\infty_{24}, 57_1, 11_1, 13_0\}  &21_{0}  &\{\infty_{25}, 16_1,  5_0,  7_0\}    &18_{0}\\
  \{\infty_{26}, 58_1,  4_0, 57_0\}    & 5_{0}   &\{\infty_{27}, 41_1, 13_1, 31_1\}  &19_{0}  &\{\infty_{28}, 64_1, 56_0, 30_1\}    &30_{0}\\
  \{\infty_{29}, 19_0, 48_0, 21_0\}    & 6_{0}   &\{\infty_{30}, 48_1, 58_0, 50_0\}  &33_{0}  &\{\infty_{31}, 40_0, 49_1,  5_1\}    &34_{0}\\
  \hline
  \end{array}
\end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
 {\cal C}= &\{ 2_1,  3_1, 22_0, 69_0\}  ,&\{28_1, 69_0, 19_1, 62_0\}  ,&\{41_1,  4_0, 20_1, 59_0\},\\
   &\{57_0, 12_1,  4_0, 55_1\}  ,&\{41_0, 21_1, 32_1,  8_0\}  ,&\{ 7_1, 13_0, 14_1, 28_0\},\\
   &\{33_1, 21_0, 28_1, 52_0\}.\\
 {\cal R}=&\{66_1,  3_1, 25_0, 29_1\}  ,&\{38_0, 34_0,  3_0, 24_0\} ,&\{55_0, 15_0, 62_0, 45_0\},\\
   &\{62_1, 61_0, 42_0, 29_0\}  ,&\{51_0, 35_0, 30_0, 26_0\}  ,&\{61_1,  1_1, 14_0, 38_1\},\\
   &\{14_1, 11_0, 31_0, 63_0\}  ,&\{ 7_1, 33_0,  8_1, 41_0\}.\\
 \end{array}
 \end{equation*}
 }
 \qed

 \begin{lemma}\label{IHGBTD class3}
 For each $(u,w)\in\{(37,6),(38,7),$ $(39,6)\}$, an IHGBTD$\left(4,4^{(u,w)}\right)$ exists.
 \end{lemma}
 \noindent {\bf Proof:}
 As with Lemma \ref{IHGBTD class1}, we apply Proposition \ref{starter-adder-IHGBTD}.
 Take
 \begin{align*}
 \Gamma & = \mathbb{Z}_{u-w}\times \mathbb{Z}_2\times \mathbb{Z}_2,\\
 \Gamma_0 & = \{0\}\times \mathbb{Z}_2\times \mathbb{Z}_2\\
 \Delta &= \{((0,0,0), (1,0,0), \cdots, (u-w-1,0,0))\}, \mbox{ and }\\
 H &= \bigcup_{i=0}^{w-1}H_{i}, \mbox{ where $H_i=\{\infty_{i},\infty_{i+w},\infty_{i+2w},\infty_{i+3w}\}$ for $0\leq i\le w-1$}.
 \end{align*}
% Take $\Gamma = \mathbb{Z}_{u-w}\times \mathbb{Z}_2\times \mathbb{Z}_2$
% and $H=\bigcup_{i=0}^{w-1}H_{i}$ where $H_i=\{\infty_{i},\infty_{i+w},\infty_{i+2w},\infty_{i+3w}\}$ for $0\leq i \le w-1$.
% Then $\Gamma_0 = \{0\}\times \mathbb{Z}_2\times \mathbb{Z}_2$ is a subgroup of order four in $\Gamma$ and
% %Take the representative system of the cosets of $G_0$
% take $\Delta$ to be $((0,0,0), (1,0,0), \cdots, (u-w-1,0,0))$.
 The desired intransitive starter and corresponding adder for $(u,w)\in\{(37,6),(38,7),$ $(39,6)\}$ are displayed below.
 Here we write the element $(a,b,c)$ of $\Gamma$ as $a_{bc}$ for succinctness.

When $(u,w)=(37,6)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
% $$
% {\footnotesize
% \begin{array}{ccccccc}
% &&&(u,w)=(37,6)&&&\\
% \hline
%      &S &A(S) &S &A(S) &S &A(S)\\
%  \hline
  \{ 6_{00}, 25_{00},  3_{00},  7_{11}\}        &30_{00}   &\{20_{10}, 13_{00}, 23_{11}, 27_{01}\}      &28_{00}  &\{12_{00}, 13_{01}, 19_{11}, 17_{00}\}        & 2_{00}\\
  \{20_{11}, 19_{00},  9_{00},  1_{11}\}        &17_{00}   &\{29_{11}, 26_{11},  2_{11},  0_{01}\}      & 3_{00}  &\{21_{10}, 11_{10},  1_{10}, 27_{10}\}        &21_{00}\\
  \{ 9_{01}, 27_{11},  4_{10}, 16_{11}\}        &11_{00}   &\{\infty_{0},  26_{01}, 28_{01},  5_{00}\}  & 4_{00}  &\{\infty_{1},  14_{10},  3_{11}, 25_{11}\}    &29_{00}\\
  \{\infty_{2},  21_{00}, 11_{11}, 23_{01}\}    &24_{00}   &\{\infty_{3},  21_{11},  5_{10}, 18_{00}\}  & 7_{00}  &\{\infty_{4},  28_{11}, 10_{11}, 20_{01}\}    & 0_{00}\\
  \{\infty_{5},  28_{10}, 25_{01}, 15_{11}\}    &25_{00}   &\{\infty_{6},   0_{10},  2_{01},  7_{10}\}  &14_{00}  &\{\infty_{7},  29_{01}, 10_{10}, 22_{00}\}    &12_{00}\\
  \{\infty_{8},   3_{01}, 12_{11}, 19_{10}\}    & 8_{00}   &\{\infty_{9},  30_{01}, 27_{00},  8_{11}\}  &27_{00}  &\{\infty_{10}, 19_{01}, 21_{01},  2_{00}\}    &23_{00}\\
  \{\infty_{11},  4_{11}, 22_{11},  7_{00}\}    &20_{00}   &\{\infty_{12}, 26_{00},  6_{01},  4_{00}\}  &19_{00}  &\{\infty_{13}, 28_{00}, 22_{01}, 14_{01}\}    &22_{00}\\
  \{\infty_{14},  2_{10}, 16_{01}, 22_{10}\}    &13_{00}   &\{\infty_{15},  4_{01}, 29_{00},  7_{01}\}  &18_{00}  &\{\infty_{16}, 24_{00},  8_{01},  5_{11}\}    &16_{00}\\
  \{\infty_{17}, 18_{11},  1_{01}, 15_{10}\}    & 1_{00}   &\{\infty_{18}, 17_{01}, 23_{10},  8_{00}\}  &26_{00}  &\{\infty_{19}, 24_{10}, 16_{00},  8_{10}\}    &10_{00}\\
  \{\infty_{20},  3_{10}, 18_{01}, 24_{01}\}    & 5_{00}   &\{\infty_{21}, 30_{11}, 24_{11}, 18_{10}\}  & 9_{00}  &\{\infty_{22},  0_{11}, 14_{11}, 23_{00}\}    &15_{00}\\
  \{\infty_{23},  6_{10}, 15_{01}, 29_{10}\}    & 6_{00} \\
  \hline
  \end{array}
\end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
 {\cal C}= &\{30_{10}, 13_{00},  7_{11},  8_{01}\}, & \{ 7_{01},  2_{10}, 28_{11}, 17_{00}\}  ,  & \{ 6_{11},  9_{01}, 10_{00}, 13_{10}\},\\
  &\{30_{10}, 28_{01}, 18_{00}, 17_{11}\}  ,  & \{30_{01}, 26_{00},  8_{11},  6_{10}\}.\\

{\cal R}=&\{14_{00}, 30_{00}, 13_{10},  0_{00}\}  ,  & \{ 9_{10}, 16_{10}, 15_{00}, 11_{00}\}  , & \{ 10_{00}, 25_{10}, 17_{10}, 30_{10}\},\\
  &\{20_{00},  5_{01},  9_{11},  1_{00}\}  ,  & \{26_{10}, 12_{10}, 13_{11}, 17_{11} \}  , & \{12_{01}, 11_{01}, 10_{01},  6_{11}\}.\\

  \end{array}
  \end{equation*}
 }

 \vskip 5pt
 When $(u,w)=(38,7)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
% $$
% {\footnotesize
% \begin{array}{ccccccc}
% &&&(u,w)=(38,7)&&&\\
% \hline
%      &S &A(S) &S &A(S) &S &A(S)\\
% \hline
  \{    28_{00}, 29_{00}, 22_{11}, 27_{00}\}    & 8_{00}   &\{20_{11}, 23_{11}, 11_{11},  5_{11}\}      & 6_{00}  &\{    18_{00}, 27_{10},  8_{01}, 30_{00}\}    &21_{00}\\
  \{\infty_{0},  30_{01}, 13_{00},  5_{01}\}    & 3_{00}   &\{\infty_{1},  28_{01},  3_{01}, 23_{01}\}  &20_{00}  &\{\infty_{2},  27_{11},  8_{10}, 24_{11}\}    &25_{00}\\
  \{\infty_{3},   0_{11},  4_{11},  6_{00}\}    &11_{00}   &\{\infty_{4},   4_{00},  9_{00},  8_{00}\}  &26_{00}  &\{\infty_{5},  16_{11}, 29_{10}, 10_{01}\}    &12_{00}\\
  \{\infty_{6},  26_{00}, 29_{01}, 21_{01}\}    & 0_{00}   &\{\infty_{7},  27_{01}, 16_{00}, 18_{10}\}  &19_{00}  &\{\infty_{8},   7_{01}, 23_{00}, 13_{11}\}    & 1_{00}\\
  \{\infty_{9},  30_{11},  6_{10}, 16_{10}\}    &28_{00}   &\{\infty_{10}, 13_{01}, 24_{10}, 22_{00}\}  &14_{00}  &\{\infty_{11},  2_{00}, 20_{00}, 12_{11}\}    &13_{00}\\
  \{\infty_{12}, 11_{00}, 23_{10}, 12_{10}\}    &16_{00}   &\{\infty_{13},  1_{10}, 15_{00}, 14_{11}\}  &18_{00}  &\{\infty_{14}, 18_{11}, 10_{10}, 12_{01}\}    &22_{00}\\
  \{\infty_{15},  3_{00}, 25_{00}, 17_{00}\}    &27_{00}   &\{\infty_{16}, 12_{00}, 26_{11}, 22_{10}\}  &29_{00}  &\{\infty_{17},  1_{01}, 17_{01}, 10_{00}\}    & 9_{00}\\
  \{\infty_{18},  0_{00}, 19_{11}, 20_{10}\}    &23_{00}   &\{\infty_{19}, 24_{00},  2_{11},  4_{10}\}  &10_{00}  &\{\infty_{20},  5_{00},  2_{10},  1_{11}\}    &17_{00}\\
  \{\infty_{21}, 25_{10},  7_{10},  0_{01}\}    &15_{00}   &\{\infty_{22}, 17_{10}, 20_{01}, 19_{10}\}  &30_{00}  &\{\infty_{23}, 14_{00}, 21_{11},  7_{00}\}    & 7_{00}\\
  \{\infty_{24},  0_{10},  4_{01}, 11_{01}\}    & 5_{00}   &\{\infty_{25},  9_{11}, 19_{01}, 21_{10}\}  & 4_{00}  &\{\infty_{26},  9_{01}, 24_{01}, 25_{11}\}    & 2_{00}\\
  \{\infty_{27}, 14_{01}, 25_{01}, 30_{10}\}    &24_{00}  \\
  \hline
 \end{array}
\end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
 {\cal C}=&\{14_{00}, 29_{11}, 25_{01}, 30_{10}\}, &\{20_{10},  9_{11},  7_{01},  5_{00}\}   , &\{ 4_{01}, 25_{00}, 28_{11}, 12_{10}\}, \\
   &\{13_{00}, 24_{10},  1_{01}, 22_{11}\}   , &\{ 7_{10},  6_{01}, 20_{11}, 10_{00}\}   , &\{24_{01},  6_{10},  1_{00}, 16_{11}\}.\\

 {\cal R}=&\{ 8_{11},  5_{10}, 19_{00}, 15_{10}\}   , &\{26_{01},  7_{11}, 13_{10}, 17_{11}\}   , &\{ 9_{10}, 15_{11},  6_{01},  1_{00}\}, \\
   &\{26_{10}, 14_{10}, 21_{00}, 28_{10}\}   , &\{22_{01}, 18_{01}, 10_{11}, 15_{01}\}   , &\{ 3_{11},  2_{01}, 16_{01}, 29_{11}\}, \\
   &\{ 3_{10}, 28_{11}, 11_{10},  6_{11}\}.\\

 \end{array}
 \end{equation*}
 }

 \vskip 5pt
 When $(u,w)=(39,6)$:
 {\footnotesize
 \begin{equation*}
 \begin{array}{lclclc}
 \hline
  {\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S}) &{\cal S} &\Omega({\cal S})\\
 \hline
%  $$
% {\footnotesize
% \begin{array}{ccccccc}
% &&&(u,w)=(39,6)&&&\\
% \hline
%      &S &A(S) &S &A(S) &S &A(S)\\
% \hline
  \{28_{10}, 29_{10}, 26_{10},  2_{00}\}        &23_{00}   &\{24_{01}, 10_{11},  9_{01}, 17_{00}\}  &13_{00}  &\{ 3_{00}, 29_{00},  6_{00}, 21_{01}\}    & 0_{00}\\
  \{11_{01}, 30_{01}, 10_{00},  7_{11}\}        &10_{00}   &\{ 9_{11}, 26_{00}, 21_{00}, 20_{01}\}  &11_{00}  &\{30_{00}, 32_{00},  0_{01},  8_{00}\}    & 8_{00}\\
  \{22_{01},  8_{10}, 18_{00}, 27_{01}\}        & 9_{00}   &\{21_{10}, 30_{11}, 24_{00},  4_{11}\}  & 5_{00}  &\{32_{01}, 27_{10}, 18_{01}, 25_{00}\}    &25_{00}\\
  \{\infty_{0},  28_{01}, 16_{00}, 12_{11}\}    &32_{00}   &\{\infty_{1},   1_{01}, 18_{10}, 16_{01}\}  &20_{00}  &\{\infty_{2},   9_{10},  6_{11},  4_{01}\}    & 3_{00}\\
  \{\infty_{3},  15_{00}, 32_{10},  6_{10}\}    &19_{00}   &\{\infty_{4},  32_{11}, 30_{10},  1_{10}\}  &27_{00}  &\{\infty_{5},  29_{01},  8_{11}, 31_{00}\}    &16_{00}\\
  \{\infty_{6},  26_{01}, 14_{11}, 23_{00}\}    &18_{00}   &\{\infty_{7},  28_{00}, 13_{01}, 24_{10}\}  &15_{00}  &\{\infty_{8},  24_{11}, 31_{01}, 13_{10}\}    &31_{00}\\
  \{\infty_{9},  27_{00}, 18_{11}, 12_{10}\}    &28_{00}   &\{\infty_{10}, 25_{11}, 13_{11}, 19_{11}\}  &22_{00}  &\{\infty_{11},  5_{10},  4_{00},  0_{00}\}    &30_{00}\\
  \{\infty_{12},  7_{00}, 13_{00}, 19_{01}\}    & 6_{00}   &\{\infty_{13},  2_{10}, 16_{11}, 25_{01}\}  &26_{00}  &\{\infty_{14}, 17_{01},  7_{01}, 11_{10}\}    & 7_{00}\\
  \{\infty_{15}, 15_{01}, 19_{10},  2_{11}\}    &17_{00}   &\{\infty_{16}, 22_{00}, 12_{00},  1_{00}\}  & 4_{00}  &\{\infty_{17},  0_{10}, 14_{01},  5_{00}\}    & 1_{00}\\
  \{\infty_{18}, 15_{11},  2_{01}, 14_{00}\}    &12_{00}   &\{\infty_{19},  4_{10},  3_{01}, 23_{11}\}  & 2_{00}  &\{\infty_{20},  3_{10}, 16_{10}, 17_{10}\}    &14_{00}\\
  \{\infty_{21},  3_{11}, 19_{00}, 25_{10}\}    &29_{00}   &\{\infty_{22},  5_{11}, 11_{00}, 22_{11}\}  &24_{00}  &\{\infty_{23}, 10_{10}, 22_{10}, 23_{01}\}    &21_{00}\\
   \hline
 \end{array}
\end{equation*}
 \begin{equation*}
 \begin{array}{rlll}
 {\cal C}= &\{10_{11}, 15_{10}, 23_{00}, 13_{01}\},  &\{22_{11},  4_{01}, 20_{00}, 27_{10}\},   &\{12_{10}, 16_{11},  8_{00},  4_{01}\},  \\
   &\{23_{11}, 12_{01},  1_{00},  9_{10}\},   &\{20_{00}, 30_{01}, 23_{10}, 28_{11}\}.\\

{\cal R}= &\{20_{11},  6_{01}, 28_{11},  5_{01}\} ,  &\{29_{11}, 12_{01}, 11_{11}, 31_{11}\} ,  &\{31_{10}, 10_{01}, 15_{10},  7_{10}\},  \\
   &\{ 9_{00}, 27_{11}, 14_{10}, 20_{00}\} ,  &\{23_{10},  0_{11}, 20_{10},  8_{01}\} ,  &\{26_{11},  1_{11}, 21_{11}, 17_{11}\}.  \\
\end{array}
\end{equation*}
}
 \qed

%Finally, we prove Theorem \ref{main th}.

%\begin{proof}
{\bf Proof of Theorem \ref{main th}}: We first construct a
GBTD$(4,m)$ for any $m\in N$, where $N= \{28, 32, 33, 34,37, 38, 39,
44\}$.

For each $w\in\{5,6,7,8\}$, an HGBTD$(4,4^w)$ is given by Yin {\em
et al.} \cite{YYW}.
 For each $m\in N$, apply Theorem \ref{IHGBTD fill in hole}, with IHGBTDs from
 Lemma \ref{IHGBTD class1}, Lemma \ref{IHGBTD class2} and Lemma \ref{IHGBTD class3}
 and corresponding HGBTD$(4,4^w)$'s where $w\in \{5,6,7,8\}$ as ingredients,
 to produce the desired HGBTD$(4,4^m)$.
 Hence, the desired GBTD$(4,m)$ follows from Proposition \ref{HGBTD+GBTD coro}.

 Combining Proposition \ref{prop:2}, Proposition \ref{prop:3} and Proposition \ref{prop:4},
 we complete the proof. \qed
%
% Now we are in the position to present our main result, Theorem \ref{main th}.
% %\begin{theorem}\label{main th}
%% A GBTD$(4,m)$ exists if and only if $m\geq 1$ and $m\neq 2,3$.
%% \end{theorem}
%
% \begin{proof} {\bf of Theorem \ref{main th}:} A GBTD$(4,1)$ exists trivially. When $m=2$ and $3$, a
% GBTD$(4,m)$ does not exist by exhaustive search with the aid of a computer.
%
% When $m=4$, we construct a GBTD$(4,m)$ on $\mathbb{Z}_4\times \mathbb{Z}_4$ as
% follows. The point $(a,b)\in \mathbb{Z}_4\times \mathbb{Z}_4$ is written as $ab$. We
% present the following four arrays $A_0,A_1,A_2$ and $A_3$.
%  $$
% A_0=
% \begin{array}{|ccc|cccc}
% \hline
% \{00,  02,  20,  22\}&\{11,  13,  31,  33\}&\{10,  12,  30,  32\}\\
% \{01,  03,  21,  23\}&\{00,  02,  20,  22\}&\{11,  13,  31,  33\}\\
% \{10,  12,  30,  32\}&\{01,  03,  21,  23\}&\{00,  02,  20,  22\}\\
% \{11,  13,  31,  33\}&\{10,  12,  30,  32\}&\{01,  03,  21,  23\}\\
% \hline
% \end{array}
% $$
% $$
% A_1=
% \begin{array}{|cccc|ccc}
% \hline
% \{23,  22,  32,  11\}&\{20,  23,  33,  12\}&\{21,  20,  30,  13\}&\{22,  21,  31,  10\}\\
% \{20,  01,  30,  33\}&\{21,  02,  31,  30\}&\{22,  03,  32,  31\}&\{23,  00,  33,  32\}\\
% \{31,  00,  12,  21\}&\{32,  01,  13,  22\}&\{33,  02,  10,  23\}&\{30,  03,  11,  20\}\\
% \{02,  10,  13,  03\}&\{03,  11,  10,  00\}&\{00,  12,  11,  01\}&\{01,  13,  12,  02\}\\
% \hline
% \end{array}
% $$
% $$
% A_2=
% \begin{array}{|cccc|ccc}
% \hline
% \{10,  00,  21,  11\}&\{11,  01,  22,  12\}&\{12,  02,  23,  13\}&\{13,  03,  20,  10\}\\
% \{33,  02,  03,  12\}&\{30,  03,  00,  13\}&\{31,  00,  01,  10\}&\{32,  01,  02,  11\}\\
% \{01,  13,  20,  32\}&\{02,  10,  21,  33\}&\{03,  11,  22,  30\}&\{00,  12,  23,  31\}\\
% \{22,  23,  30,  31\}&\{23,  20,  31,  32\}&\{20,  21,  32,  33\}&\{21,  22,  33,  30\}\\
% \hline
% \end{array}
% $$
%  $$
% A_3=
% \begin{array}{|cccc|ccc}
% \hline
% \{00,  01,  30,  33\}&\{01,  02,  31,  30\}&\{02,  03,  32,  31\}&\{03,  00,  33,  32\}\\
% \{10,  13,  22,  23\}&\{11,  10,  23,  20\}&\{12,  11,  20,  21\}&\{13,  12,  21,  22\}\\
% \{02,  11,  21,  32\}&\{03,  12,  22,  33\}&\{00,  13,  23,  30\}&\{01,  10,  20,  31\}\\
% \{03,  12,  20,  31\}&\{00,  13,  21,  32\}&\{01,  10,  22,  33\}&\{02,  11,  23,  30\}\\
% \hline
% \end{array}
% $$
%
%  It is readily checked the following array $R$ forms a
%  GBTD$(4,4)$.
%  $$
%  R=
%  \begin{array}{|c|c|c|c|ccc}
%  \hline
%  A_0&A_1&A_2&A_3\\
%  \hline
%  \end{array}
%  $$
%
% When $m\geq 5$, let $N=\{28, 32, 33, 34,37, 38, 39,
% 44\}$.  Then we only need to construct a GBTD$(4,m)$ for any $m\in N$ by Theorem \ref{old GBTD}.
% For each $w\in\{5,6,7,8\}$, an HGBTD$(4,4^w)$ is given in \cite{YYW}.
% For each $m\in N$, apply Theorem \ref{IHGBTD fill in hole}, with IHGBTDs from Lemma \ref{IHGBTD class1}-Lemma \ref{IHGBTD class3}
% and corresponding HGBTD$(4,4^w)$'s where $w\in \{5,6,7,8\}$ as ingredients,
% to produce an HGBTD$(4,4^m)$. Then
% the result follows by Proposition \ref{HGBTD+GBTD coro}.
% \end{proof}



 \vspace{0.3cm} \noindent{\bf Acknowledgement}
 We are grateful to the anonymous reviewers for their helpful
 comments.

% The research is supported by NSFC under Grants No. 10801064 and
% 11001109, and by the Program for Innovative Research Team of
% Jiangnan University.

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\end{document}
