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\title{\bf The existence of NGBTDs and its application}

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\author{Chengmin Wang, Jie Yan\footnote{Corresponding
author. Email: jyan7906@yahoo.com.cn}\\
\small School of Science, Jiangnan University,\\[-0.8ex]
\small Wuxi 214122, China \\
}

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% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{XXX, XXX}{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
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\begin{abstract}
  In this paper, we complete the existence of near generalized
 balanced tournament designs (NGBTDs) with block size 3 and 5.
 As its application, we obtain new classes of optimal constant composition codes.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} near generalized balanced tournament
 design; constant composition codes; optimal; existence
\end{abstract}

\section{Introduction}
 Let $X$ be a set of $v$ {\em points}, and ${\cal A}$ a collection of
 subsets (called {\em blocks}) of $X$. A $(v, k, \lambda)$ {\em balanced incomplete block
 design} (BIBD), or a $(v, k, \lambda)$-BIBD, is a pair $(X,{\cal A})$ such that any pair of distinct points of $X$ occurs
 in precisely $\lambda$ blocks.
 A $(km+1, k, k - 1)$-BIBD $(X, {\cal A})$ is called a {\em near generalized balanced tournament
 design} (NGBTD), or an NGBTD$(k,m)$ in short, if its blocks
 can be arranged into an $m\times (km+1)$ array in such a way
 that

 (1) the blocks in each column form a partial parallel class which partition $X\backslash \{x\}$ for some point $x\in X$;

 (2) each point of $X$ is contained in precisely $k$ cells of each
 row.\\
 By the definition, any NGBTD can be identified with its
 corresponding block array defined above.

 NGBTDs are a generalization of odd
 balanced tournament designs (OBTDs). Particularly, an NGBTD$(2, m)$ is an
 OBTD$(m)$  whose existence was completed in \cite{LV}. Lamken \cite{Lamken} almost finished the existence of NGBTD$(3,
 m)$ with four possible exceptions.
 Recently, Shan \cite{Shan} presented a nearly complete solution for the existence of NGBTD$(k, m)$'s for $k=4$ and 5 with four possible exceptions.
 We collect the existence of NGBTD$(k,
 m)$'s for $2\leq k \leq5$ as follows.

 \begin{theorem}\label{old NGBTD}{\rm (\cite{Lamken,LV,Shan})}\\
  {\rm (1)} There exists an NGBTD$(2,m)$ for any positive integer $m$;\\
  {\rm (2)} There exists an NGBTD$(3,m)$ for any positive integer $m$ and $m\not\in \{3, 38, 39,
 118\}$;\\
  {\rm (3)} There exists an NGBTD$(4,m)$ for any positive integer $m$;\\
  {\rm (4)} There exists an NGBTD$(5,m)$ for any positive integer $m$ and $m\not\in \{15,32,40,45\}$.
 \end{theorem}

 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 $g_1^{u_1}g_2^{u_2}\cdots g_s^{u_s}$ means $u_i$ occurrences of $g_i$ for $1\leq i\leq
 s$.

 A set of blocks of a GDD $(X, {\cal G}, {\cal B})$ is called a {\em partial $\alpha$-parallel
 class} over $X\backslash S$ 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 called {\em resolvable} if its blocks can be
 partitioned into parallel classes.

 We need a special type of GDDs, called {\em frame
 generalized doubly resolvable packings} (FGDRPs), which was
 first introduced in \cite{YY1} to construct optimal constant composition codes.


 Let $(X, {\cal G}, {\cal B})$ be a  $(k,k-1)$-GDD of type $g^u$ where $g=kh$.
 Suppose that the group set ${\cal G}=\{G_1,G_2,\cdots, G_{u}\}$.
 Define
 {\small
\begin{center}
\tabcolsep 0.3 cm
            \begin{tabular}{l}
            $R_i=\{(i-1)h+j:j=1,\cdots, h\}$\\
            $C_i=\{(i-1)kh+j:j=1,\cdots, kh\}$
             \end{tabular}
 \end{center}}
  \noindent for $1\leq i\leq u$.
 The blocks of the GDD can be arranged into an $hu\times gu$
 array $F$ which satisfies the following properties:

 \noindent
 (1) Each cell of $F$ is either empty or contains a block of ${\cal
 B}$;\\
 (2) Let $F_t$ be the subarray indexed by the elements of $R_t$ and $C_t$. Then $F_t$ is empty for
 $t = 1, 2, \cdots, u$, i.e., the main diagonal of $F$ consists of $u$ empty
 subarrays of size $h\times kh$;\\
 (3) For any $x \in R_i$ $(1\leq i\leq u)$, the blocks in row $x$ form a partial $k$-parallel class over
 $X\backslash G_i$;\\
 (4) For any $y\in C_j$ $(1\leq j\leq u)$, the blocks in column $y$ form a partial parallel class over $X\backslash
 G_j$.

 Then we refer to this GDD as an FGDRP$(k, g^u)$.
 Actually, FGDRPs can be defined in a more general way in which all the groups are not
 necessarily the same size. Nevertheless, we use the definition here for
 our purpose. The interest reader may refer to
 \cite{YW,YY1} for more details on FGDRPs.

 In this note, we completely establish the existence of NGBTD$(k,m)$'s for $k=3$ and 5 by removing all the remaining cases
 in Theorem \ref{old NGBTD}.
 We also present an application of NGBTDs to optimal constant
 composition codes.



 \section{The Existence of NGBTDs with Block Size 3}
 In this section, we give a complete solution to the existence of NGBTDs with block size
 3.

 \begin{theorem}\label{FGBTD to NGBTD}{\rm \cite{Shan,Yan}}
 Suppose that there exists an FGDRP$(k,g^u)$ where $g=kh$.
 Let $w$ be any nonnegative integer. If there exists an
 NGBTD$(k,h+w)$ which contains an NGBTD$(k,w)$ as a subarray, then an NGBTD$(k,hu+w)$
 exists.
 \end{theorem}

 \begin{theorem}\label{FGBTD(3,g^u)} {\rm \cite{Yan,YW}}
 Let $g$ and $u$ be positive integers with $g\equiv 0$ $(mod\ 3)$ and $u\geq 5$. Then
 an FGDRP$(3, g^u)$ exists except possibly for $(g, u)\in \{(6, 15), (9, 18), (9, 28), (9, 34),(30, 15)\}$.
 \end{theorem}

 \begin{theorem}\label{NGBTD(3,m)}
 For any integer $m\geq 1$ and $m\neq 3$, there exists an NGBTD$(3, m)$.
 There does not exist an NGBTD$(3, 3)$.
 \end{theorem}
 $Proof:$
 By Theorem \ref{FGBTD(3,g^u)}, there exists an FGDRP$(3, 3^m)$ for any $m\geq
 5$. An NGBTD$(3,1)$ exists trivially with all the subsets containing any three points as blocks.
 Then we apply Theorem \ref{FGBTD to NGBTD} with $k=3,h=1,u=m,w=0$ to
 obtain an NGBTD$(3,m)$ for any $m\geq 5$.

 We know that there does not exist an NGBTD$(3, 3)$ definitely by an exhaustive search with the aid of a computer.

 Combining with Theorem \ref{old NGBTD}, we complete the proof.
 \qed

 \noindent
 {\bf Remark:} Here we have fixed the four possible exceptions in \cite {Lamken} and complete the existence of NGBTD$(3,m)$'s.
 Moreover, Theorem \ref{NGBTD(3,m)} actually provides an alternative proof
 for the existence of NGBTD$(3, m)$'s.

 \section{The Existence of NGBTDs with Block Size 5}
 This section serves to complete the existence of NGBTDs with block size
 5, by removing the four outstanding cases in Theorem \ref{old NGBTD}.
 \begin{lemma}\label{$NGBTD(5,15)$}
 There exists an NGBTD$(5,15)$.
 \end{lemma}
 $Proof:$  We construct the desired NGBTD on the Abelian group $X=\mathbb{Z}_{19}\times \mathbb{Z}_2\times
 \mathbb{Z}_2$. Here we write the element $(a,b,c)$ of $X$ as $\underline{a}bc$ for brevity.
 The following blocks form a partial parallel class which partitions $X\backslash \{\underline{18}11\}$.
 {\small
 $$
 \begin{array}{llllllllllll}
 \{\underline{ 0}01,\underline{ 0}00,\underline{ 0}10,\underline{ 0}11,\underline{ 1}00\}&\{\underline{16}01,\underline{ 3}01,\underline{12}10,\underline{ 9}10,\underline{14}01\}&\{\underline{14}11,\underline{13}11,\underline{ 7}01,\underline{17}01,\underline{ 2}00\}\\
 \{\underline{15}01,\underline{12}00,\underline{18}01,\underline{ 3}11,\underline{11}01 \}&\{\underline{14}00,\underline{ 9}11,\underline{13}00,\underline{12}11,\underline{ 8}10\}&\{\underline{13}01,\underline{15}00,\underline{ 4}00,\underline{11}11,\underline{ 2}10 \}\\
 \{\underline{ 1}01,\underline{ 3}00,\underline{10}11,\underline{11}00,\underline{17}11\}&\{\underline{ 1}11,\underline{ 4}01,\underline{ 6}11,\underline{ 9}01,\underline{14}10 \}&\{\underline{ 3}10,\underline{16}11,\underline{ 7}11,\underline{10}00,\underline{ 5}01\}\\
 \{\underline{ 5}00,\underline{ 6}10,\underline{12}01,\underline{ 2}11,\underline{17}00 \}&\{\underline{ 4}11,\underline{ 7}00,\underline{ 1}10,\underline{16}10,\underline{ 8}11\}&\{\underline{ 5}10,\underline{10}10,\underline{ 6}00,\underline{ 7}10,\underline{11}10\}\\
 \{\underline{ 6}01,\underline{ 4}10,\underline{15}10,\underline{ 8}00,\underline{17}10\}&\{\underline{18}00,\underline{16}00,\underline{ 9}00,\underline{ 8}01,\underline{ 2}01
 \}& \{\underline{18}10,\underline{15}11,\underline{13}10,\underline{10}01,\underline{ 5}11\}\\
 \end{array}
 $$
 }
 It is easy to check that the desired design is produced by the
 action of the group $X$ on above partial parallel class.
 \qed

 Before we move to the next construction, we need the well-known
 starter-adder method, which is widely used to produce some designs
 with orthogonal properties (see, for example, \cite{CD,CLLM,LV,RV,Shan,YYW}).

 Let $G$ be an additively Abelian group of order $u$. Let $g=kt$. A
 {\em starter} $S$ for an FGDRP$(k,g^u)$ over $\mathbb{Z}_g\times G$ with groups $G_x=\mathbb{Z}_g\times
 \{x\}$ $(x\in G)$ consists of $t$ sets of $k$-tuples (base blocks),
 $S_1,S_2,\cdots, S_t$, which satisfies the following properties:


 (1) For any $i$ $(1\leq i\leq t)$, $S_i$ contains exactly $u-1$ base blocks $B_{ij}$, $j=1,2,\cdots, u-1$;

 (2) $S=\bigcup_{i=1}^{t} S_i$ forms a partition of $\mathbb{Z}_g\times (G\backslash \{0\})$ and
 every element of $\mathbb{Z}_g\times (G\backslash \{0\})$ occurs exactly
 $k-1$ times in the difference list of $S$.

 A corresponding {\em adder} $A(S)$ for the starter $S$ consists $t$
 permutations (not necessarily distinct) of $G\backslash \{0\}$
 $$
    A(S_i)=(a_{i1},a_{i2}, \cdots, a_{i(u-1)}), 1\leq i\leq t
 $$ such that for any $i$ $(1\leq i\leq t)$, $\bigcup_{j=1}^{u-1}
 (B_{ij}+(0,a_{ij}))$ contains exactly $k$ elements (not necessarily distinct)
 from each group $G_x$ for $x\in G\backslash \{0\}$ and no element of
 $G_0$.

 \begin{theorem}\label{starter-adder-type1}{\rm{\cite {YW}}}
 If there exists a starter-adder pair $(S,A(S))$ for an
 FGDRP$(k,g^u)$ over $\mathbb{Z}_g\times G$ defined above, then there exists an
 FGDRP$(k,g^u)$.
 \end{theorem}
 \begin{lemma}\label{$FGDRP(25^9)$}
 There exists an FGDRP$(5,25^9)$.
 \end{lemma}
 $Proof:$ We take $\mathbb{Z}_{25}\times GF(9)$ as the
 point set and $\{\mathbb{Z}_{25}\times
 \{x\}\mid x\in GF(9)\}$ as the groups. Let $GF(9)^*=GF(9)\backslash\{0\}$.
 Suppose that ${\omega}$ is a primitive element of $GF(9)$ satisfying ${\omega}^2={\omega}+1$ and $C_0^2$ is the square residue of
 $GF(9)^*$. We display the starter $S$ and the corresponding adder $A(S)$ in Table \ref{25^9}.
 Then we apply Theorem \ref{starter-adder-type1} to produce the desired design.
 \begin{table}[htbp]
 \begin{center}
 \caption{The starters and corresponding adders for an $FGDRP(25^9)$}\label{25^9}
 {\small
 \begin{tabular}{|ccc|c|}
 \hline
      &$S$ &$A(S)$&\\
 \hline
 $S_1:$ &\{(0,${\omega}^1$), (8,${\omega}^5$), (8,${\omega}^2$), (13,${\omega}^3$), (14,${\omega}^7$)\}$\cdot(1,h)$   &(0,1)$\cdot(1,h)$& \\
         &\{(9,${\omega}^0$), (10,${\omega}^7$), (21,${\omega}^1$), (23,${\omega}^2$), (15,${\omega}^6$)\}$\cdot(1,h)$ &(0,${\omega}$)$\cdot(1,h)$&\\
 $S_2:$ &\{(2,${\omega}^6$), (11,${\omega}^5$), (12,${\omega}^2$), (14,${\omega}^0$), (15,${\omega}^1$)\}$\cdot(1,h)$    &(0,1)$\cdot(1,h)$&\\
         &\{(5,${\omega}^4$), (6,${\omega}^6$), (17,${\omega}^3$), (20,${\omega}^7$), (24,${\omega}^1$)\}$\cdot(1,h)$   &(0,${\omega}$)$\cdot(1,h)$& \\
 $S_3:$ &\{(10,${\omega}^0$), (18,${\omega}^1$), (18,${\omega}^6$), (19,${\omega}^3$), (23,${\omega}^7$)\}$\cdot(1,h)$   &(0,1)$\cdot(1,h)$ &$h\in C_0^2$ \\
         &\{(2,${\omega}^1$), (4,${\omega}^3$), (4,${\omega}^6$), (11,${\omega}^2$), (13,${\omega}^4$)\}$\cdot(1,h)$ &(0,${\omega}$)$\cdot(1,h)$&\\
 $S_4:$ &\{(3,${\omega}^6$), (7,${\omega}^1$), (19,${\omega}^0$), (22,${\omega}^2$), (22,${\omega}^7$)\}$\cdot(1,h)$    &(0,1)$\cdot(1,h)$&\\
         &\{(1,${\omega}^2$), (6,${\omega}^1$), (9,${\omega}^7$), (16,${\omega}^0$), (24,${\omega}^6$)\}$\cdot(1,h)$   &(0,${\omega}$)$\cdot(1,h)$&\\
 $S_5:$ &\{(1,${\omega}^1$), (12,${\omega}^5$), (16,${\omega}^3$), (17,${\omega}^6$), (20,${\omega}^0$)\}$\cdot(1,h)$   &(0,1)$\cdot(1,h)$ &\\
         &\{(0,${\omega}^0$), (3,${\omega}^1$), (5,${\omega}^3$), (7,${\omega}^2$), (21,${\omega}^4$)\}$\cdot(1,h)$ &(0,${\omega}$)$\cdot(1,h)$ &\\
 \hline
 \end{tabular}
 }
\end{center}
\end{table}
\qed

 The second type of starter-adder construction is called intransitive starter-adder method, which involves infinite
 points. The reader may refer to \cite{CLLM,LV,RV,Shan,YW,YYW} for more details.

 Let $GF(u-1)$ be the Galois field with $u-1$ elements and $X=\mathbb{Z}_g\times$ $(GF{(u-1)}\bigcup
 \{\infty\})$ where $g=kt$. Let $G=\mathbb{Z}_g\times GF{(u-1)}$.
 An {\em intransitive starter} $S$ for an FGDRP$(k,g^u)$
 over $X$ with groups $\{G_x=\mathbb{Z}_g\times \{x\}\mid x\in GF(u-1)\bigcup \{\infty\}\}$ is defined as a triple $(S,R,C)$ satisfying the following
 properties.

 (1) $S$ consists of $t$ sets of $k$-tuples (base blocks), $S_1,S_2,\cdots,
 S_t$. For any $i$ $(1\leq i\leq t)$, $S_i$ contains precisely $u-2$
 base blocks, $B_{ij}$ $(1\leq i\leq u-2)$, in which there exist
 exactly $k$ base blocks containing one infinite point each from
 $G_{\infty}$;

 (2) $R$ consists of $t$ base blocks over $G$, denoted by $R_1,R_2\cdots, R_t$,
 in which each contains no infinite points;

 (3) $C$ consists of $t$ base blocks over
 $G$, denoted by $C_1,C_2\cdots, C_t$, in which each contains no infinite
 points;

 (4) $S\bigcup R$ forms a partition of $X\backslash G_0$;

 (5) the difference list from the base blocks of $S\bigcup R\bigcup C$ contains every element of $G\backslash G_0$ precisely $k
 -1$ times, and no element in $G_0$.


 A corresponding {\em adder} $A(S)$ for $S$ consists of $t$ permutations on $GF(u-1)\backslash \{0\}$,
 $$
 A(S_i) = (a_{i1}, a_{i2},\cdots, a_{i(u-2)})\ (1\leq i\leq t)
 $$
 For any $i$ $(1\leq i\leq t)$, the multiset  $\bigcup_{j=1}^{u-2}
 (B_{ij}+(0,a_{ij}))\bigcup \{C_i\}$ contains exactly $k$ elements (not necessarily
 distinct) from each group $G_x$ for $x\in GF(u-1)\backslash \{0\}$ and no element
 of $G_0$.

 \begin{theorem}\label{starter-adder-type2} {\rm \cite{YW}}
 If there exists an intransitive starter $(S,R,C)$ over $X$ defined above and a corresponding
 adder $A(S)$, then there exists an FGDRP$(k,g^u)$.
 \end{theorem}

  \begin{lemma}\label{$FGDRP(20^{8})$}
 There exists an FGDRP$(5,20^{8})$.
 \end{lemma}
 $Proof:$ Let $X=\mathbb{Z}_{20} \times (\mathbb{Z}_{7}\cup
 \{\infty\})$ be the point set and $\{\mathbb{Z}_{20}\times \{x\}\mid x\in \mathbb{Z}_7\bigcup \{\infty\}\}$ the group set.
 Here we apply Theorem \ref{starter-adder-type2} with $k=5,t=4$.
 In the following table, we present $S_i,R_i,C_i$ and
 $A(S_i)$ for $i=1$ and 2.

 %\begin{table}[htbp]
 \begin{center}
 %\caption{$S_i,R_i,C_i$ and $A(S_i)$ $(i=1,2)$ for an $FGDRP(20^{8})$}\label{20^8}
 {\small
 \begin{tabular}{|cc||cc|}
 \hline
     $A(S_1)$ &$S_1$ & $A(S_2)$  &$S_2$\\
 \hline
   (0,6)   & \{(3,3), (19,6), (0,2), (17,5), (2,4)\}  & (0,6)    &\{(4,5), (17,4), (9,2), (12,3), (15,6)\} \\
   (0,5)   & \{$-$, (7,5), (14,3), (13,1), (4,6)\}    & (0,5)    &\{$-$, (15,5), (2,6), (11,3), (19,4)\}   \\
   (0,4)   & \{$-$, (2,5), (0,4), (9,6), (1,2)\}      & (0,4)    &\{$-$, (4,4), (6,6), (12,1), (13,2)\}    \\
   (0,3)   & \{$-$, (8,1), (3,6), (19,2), (10,5)\}    & (0,3)    &\{$-$, (18,3), (11,1), (12,2), (7,6)\}   \\
   (0,2)   & \{$-$, (16,2), (16,4), (1,6), (8,3)\}    & (0,2)    &\{$-$, (3,2), (5,4), (9,3), (16,6)\}     \\
   (0,1)   & \{$-$, (5,1), (5,5), (10,4), (14,2)\}    & (0,1)    &\{$-$, (6,3), (6,5), (14,1), (18,2)\}    \\
 \hline\hline
 $C_1:$&\{(4,2), (14,4), (14,3), (0,6), (0,5)\}       & $C_2:$&\{(6,3), (5,1), (8,6), (8,2), (7,5)\}     \\
 $R_1:$&\{(7,4), (11,5), (1,3), (17,1), (0,6)\}       & $R_2:$&\{(10,6), (18,1), (15,4), (13,3), (8,2)\} \\
 \hline
 \end{tabular}
 }
 \end{center}

 For $i=3$ and 4, let $S_{i}=S_{i-2}\cdot (1,-1)$, $R_{i}=R_{i-2}\cdot (1,-1)$, $C_{i}=C_{i-2}\cdot (1,-1)$ and $A(S_{i})=A(S_{i-2})\cdot
 (1,-1)$.
 Then it is easy to check that $(\cup_{i=1}^4S_i,\cup_{i=1}^4R_i,\cup_{i=1}^4C_i)$ is an intransitive starter and $A(S)=\cup_{i=1}^4A(S_i)$
 is the corresponding adder for an FGDRP$(5,20^{8})$.
 Here the twenty points
 from $\mathbb{Z}_{20}\times \{\infty\}$
 can be distributed to the five blocks of size four in each $S_i$ for $1\leq i \leq 4$ in
 an arbitrary way. So we use the symbol ``$-$" to denote any
 point from $\mathbb{Z}_{20}\times \{\infty\}$.
 \qed


 \begin{lemma}\label{$FGDRP(20^{10})$}
 There exists an FGDRP$(5,20^{10})$.
 \end{lemma}
 $Proof:$  Here we take $\mathbb{Z}_{20} \times (GF(9)\cup
 \{\infty\})$ as the point set and $\{\mathbb{Z}_{20}\times \{x\}\mid x\in GF(9)\cup \{\infty\}\}$ as the group set.
 Suppose that ${\omega}$ is a primitive element of $GF(9)$ satisfying
${\omega}^2={\omega}+1$.
 We apply Theorem \ref{starter-adder-type2} with $k=5,t=4$.
 First we display $S_1$, $R_1$, $C_1$ and $A(S_1)$ in the following table.

  \begin{center}
%  \caption{$S_1,R_1,C_1$ and $A(S_1)$ for an $FGDRP(20^{10})$}\label{20^10}
 {\small
 \begin{tabular}{|cc|}
 \hline
    $A(S_1)$  &$S_1$  \\
 \hline
  (0,$1$)       &\{(16,${\omega}^0$), (7,${\omega}^6$), (14,${\omega}^2$), (10,${\omega}^7$),  (1,${\omega}^1$)\} \\
  (0,${\omega}$)       &\{(3,${\omega}^0$), (18,${\omega}^1$), (18,${\omega}^2$), (6,${\omega}^3$),  (4,${\omega}^7$)\}  \\
  (0,${\omega}^2$)     &\{(6,${\omega}^0$), (14,${\omega}^1$), (2,${\omega}^2$), (3,${\omega}^3$),  (1,${\omega}^4$)\}   \\
  (0,${\omega}^3$)     &\{$-$, (4,${\omega}^0$), (13,${\omega}^1$), (0,${\omega}^4$), (9,${\omega}^5$)\}       \\
  (0,${\omega}^4$)     &\{$-$, (8,${\omega}^4$), (15,${\omega}^2$), (8,${\omega}^5$), (11,${\omega}^6$)\}      \\
  (0,${\omega}^5$)     &\{$-$, (12,${\omega}^5$), (16,${\omega}^3$), (15,${\omega}^7$), (17,${\omega}^6$)\}    \\
  (0,${\omega}^6$)     &\{$-$, (0,${\omega}^7$), (9,${\omega}^6$), (19,${\omega}^4$), (19,${\omega}^5$)\}      \\
  (0,${\omega}^7$)     &\{$-$, (2,${\omega}^1$), (10,${\omega}^2$), (12,${\omega}^0$), (17,${\omega}^5$)\}     \\
 \hline   \hline
 $C_1$: &\{(15,${\omega}^7$), (2,${\omega}^3$), (14,${\omega}^0$), (5,${\omega}^5$), (11,${\omega}^4$)\} \\
 $R_1$: &\{(11,${\omega}^5$), (5,${\omega}^2$), (5,${\omega}^7$), (13,${\omega}^6$), (7,${\omega}^1$)\} \\
  \hline
 \end{tabular}
 }
 \end{center}
% \end{table}

Then, for $1\leq i \leq 4$, let $S_i=S_1 \cdot
 (1,{\omega}^{2(i-1)})$, $R_i=R_1 \cdot
 (1,{\omega}^{2(i-1)})$, $C_i=C_1 \cdot
 (1,{\omega}^{2(i-1)})$ and $A(S_i)=A(S_1) \cdot
 (1,{\omega}^{2(i-1)})$.
 It is an easy matter to verify  that
 $(\cup_{i=1}^4S_i,\cup_{i=1}^4R_i,\cup_{i=1}^4C_i)$ is the required intransitive starter and  $A(S)=\cup_{i=1}^4A(S_i)$ is the
 corresponding adder.
  Similarly with Lemma \ref{$FGDRP(20^{8})$}, the twenty points
 from $\mathbb{Z}_{20}\times \{\infty\}$
 can be distributed to the five blocks of size four in each $S_i$ for $1\leq i \leq 4$ in
 an arbitrary way. So we still use the symbol ``$-$" to denote any
 point from $\mathbb{Z}_{20}\times \{\infty\}$.
\qed

 %\begin{table}[htbp]
% \begin{center}
%  \caption{$S_1,R_1,C_1$ and $A(S_1)$ for an $FGDRP(20^{10})$}\label{20^10}
% {\small
% \begin{tabular}{|cc|}
% \hline
%    $A(S_1)$  &$S_1$  \\
% \hline
%  (0,$1$)       &\{(16,${\omega}^0$), (7,${\omega}^6$), (14,${\omega}^2$), (10,${\omega}^7$),  (1,${\omega}^1$)\} \\
%  (0,${\omega}$)       &\{(3,${\omega}^0$), (18,${\omega}^1$), (18,${\omega}^2$), (6,${\omega}^3$),  (4,${\omega}^7$)\}  \\
%  (0,${\omega}^2$)     &\{(6,${\omega}^0$), (14,${\omega}^1$), (2,${\omega}^2$), (3,${\omega}^3$),  (1,${\omega}^4$)\}   \\
%  (0,${\omega}^3$)     &\{$-$, (4,${\omega}^0$), (13,${\omega}^1$), (0,${\omega}^4$), (9,${\omega}^5$)\}       \\
%  (0,${\omega}^4$)     &\{$-$, (8,${\omega}^4$), (15,${\omega}^2$), (8,${\omega}^5$), (11,${\omega}^6$)\}      \\
%  (0,${\omega}^5$)     &\{$-$, (12,${\omega}^5$), (16,${\omega}^3$), (15,${\omega}^7$), (17,${\omega}^6$)\}    \\
%  (0,${\omega}^6$)     &\{$-$, (0,${\omega}^7$), (9,${\omega}^6$), (19,${\omega}^4$), (19,${\omega}^5$)\}      \\
%  (0,${\omega}^7$)     &\{$-$, (2,${\omega}^1$), (10,${\omega}^2$), (12,${\omega}^0$), (17,${\omega}^5$)\}     \\
% \hline   \hline
% $C_1$: &\{(15,${\omega}^7$), (2,${\omega}^3$), (14,${\omega}^0$), (5,${\omega}^5$), (11,${\omega}^4$)\} \\
% $R_1$: &\{(11,${\omega}^5$), (5,${\omega}^2$), (5,${\omega}^7$), (13,${\omega}^6$), (7,${\omega}^1$)\} \\
%  \hline
% \end{tabular}
% }
% \end{center}
% \end{table}


 Now we are in a position to complete the existence of NGBTDs with
block size five.
 \begin{theorem}\label{NGBTD(5,m)}
 For any positive integer $m$, there exists an NGBTD$(5,m)$.
 \end{theorem}
 $Proof:$ By Theorem \ref{old NGBTD}, we need only to show an NGBTD$(5,m)$
 exists for each $m\in \{15,32,40,45\}$.

 An NGBTD$(5,15)$ is given in Lemma \ref{$NGBTD(5,15)$}.
 For $m\in \{32,40,45\}$, we apply Theorem \ref{FGBTD to NGBTD} with
 $k=5,w=0$ and other suitable parameters displayed in the following table to obtain the desired NGBTD$(5,m)$.
  {\small
 \begin{center}
 \tabcolsep 0.3 cm
            \begin{tabular}{ccccccccccccccccccc} \hline
               $m$  &&&$g$ && &$h$ & &  &$u$  &                & &the source of an FGDRP$(k,g^{u})$  \\\hline
                32  &&&20  && &4   & &  &8    &                & &Lemma \ref{$FGDRP(20^{8})$}    \\
                40  &&&20   && &4   & &  &10  &                & & Lemma \ref{$FGDRP(20^{10})$}  \\
                45  &&&25   && &5   & &  &9   &                & & Lemma \ref{$FGDRP(25^9)$}   \\\hline
             \end{tabular}
 \end{center}}

 \section{Applications to Constant Composition Codes}
 Let $Q =\{a_t:0\leq t\leq m-1\}$ be an arbitrary alphabet set with $m$ elements. A code $C\subseteq Q^n$ over $Q$ with size $M$ and
 minimum distance $d$ is referred to as a {\it constant composition
 code} (CCC), or an $(n, M, d, [w_0,w_1,\cdots ,w_{m-1}])_m$-CCC, if each codeword
 has precisely $w_i$ occurrences of $a_i$ for any $i$ $(0\leq i\leq m-1)$.
 Here the definition implies
 $n=\sum_{0\leq i\leq m-1}w_i$.

 Since the constant composition $[w_0,w_1,\cdots ,w_{m-1}]$ is essentially an unordered
 multiset, we usually write it in an exponential notation: a
 constant composition $[a_1^{u_1}a_2^{u_2}\cdots a_s^{u_s}]$ indicates $u_i$ occurrences of
 $a_i$ for $1\leq i\leq s$ for brevity.
 We denote the maximum size $M$ of an $(n, M, d, [w_0,w_1,\cdots ,w_{m-1}])_m$-CCC by
 $A_m(n,$ $d, [w_0,w_1,\cdots, w_{m_1}])$. A CCC with this size is called {\it optimal}. The following
 upper bound was established by Luo et al. \cite{LFHC}.

 \begin{theorem}\label{CCC Bound} {\rm{\cite{LFHC}}}
 If $nd-n^2+(w_0^2 + w_1^2 +\cdots+w_{m-1}^2)>0$, then
 $$
 A_m(n, d, [w_0,w_1,\cdots, w_{m_1}])\leq \frac{nd}{nd-n^2+(w_0^2 + w_1^2
 +\cdots+w_{m-1}^2)}.
$$
\end{theorem}

 The study of optimal CCCs has attracted  extensive attention due to
 their numerous
 applications (see, for example, \cite{DY1,LFHC} and the references therein).
 Particularly, Ding and Yin \cite{DY1} presented a combinatorial characterization of constant composition
 codes and established an equivalent relationship between CCCs and a class of designs called generalized doubly resolvable packings
 (GDRPs) which are defined below.

 Let $X$ be a set of $v$ elements (called points) and ${\cal A}$ be a collection of subsets (called blocks) of $X$.
 Then the pair $(X,{\cal A})$ is called a {\em $\lambda$-packing} of order $v$, if
 every pair of distinct points of $X$ occurs in at most $\lambda$ blocks. Furthermore, it is termed
 a {\em generalized doubly resolvable packing} (GDRP), if the blocks of ${\cal
 A}$ can be arranged into an $m\times n$ array ${\cal R}$ which satisfies the following
 properties:

 (1) Each cell of ${\cal R}$ is either empty or contains one block;

 (2) For $0\leq i\leq m-1$, the blocks in row $i$ of ${\cal R}$ form a $w_i$-parallel
 class, that is, every point occurs in exactly $w_i$ blocks;

 (3) The blocks in every column of ${\cal R}$ form a parallel class,
 that is, every point occurs in exactly one block.

 We denote such a GDRP by a GDRP$(m\times n,\lambda; v)$. The
 multiset $T = \{w_0,w_1,\cdots, w_{m-1}\}$ is called the type of the GDRP.
 For more details, the interested reader may refer to
 \cite{DY1,YY1,YY2}.

 \begin{theorem}\label{optimal CCC}{\rm {\cite{DY1,YY2}}}
 The existence of a GDRP$(m\times n,\lambda;v)$ of type $\{w_0,w_1,\cdots,w_{m-1}\}$ is equivalent
 to an $(n, M, d, [w_0,w_1,\cdots, w_{m-1}])_m$-CCC, where $M=v$ and $d=n-\lambda$.
 \end{theorem}

 \begin{theorem}\label{CCC+GDRP}
 If there exists an NGBTD$(k,m)$, then there exists an optimal
 $(km+1,km+1,k(m-1)+2,[1^1k^m])_{m+1}$-CCC.
 \end{theorem}
 $Proof:$ It is readily checked that an NGBTD$(k,m)$ is a GDRP$(m\times (km+1),k-1;km+1)$
 of type $\{1,k,\cdots, k\}$. By Theorem \ref{optimal CCC}, we have an $(n, M, d, [w_0,w_1,\cdots, w_{m-1}])_m$-CCC
 with $n=M=km+1, d=k(m-1)+2, w_0=1, w_1=w_2=\cdots=w_{m-1}=k$. In addition, by Theorem \ref{CCC Bound}, we have
 \begin{align*}
 &\ \ \ \ \ A_m(km+1, k(m-1)+2, [1^1k^m])\\
 &\ \ \ \ \ \leq  \displaystyle\frac{(km+1)(k(m-1)+2)}{(km+1)(k(m-1)+2)-(km+1)^2+(1^2 + k^2+\cdots+k^2)}\\
 &\ \ \ \ \ =\displaystyle\frac{(km+1)(k(m-1)+2)}{(km+1)(1-k)+(1+mk^2)}\\
 &\ \ \ \ \ =km+1
 \end{align*}

 Hence the obtained CCC is optimal. Then the proof is complete.
  \qed



 \begin{theorem}\label{New optimal CCC}
 Let $m,k$ be integers satisfying $m\geq 1$, $2 \leq k\leq 5$ and $(k,m)\not\in (3,3)$. Then there exists an optimal
 $(km+1,km+1,k(m-1)+2,[1^1k^m])_{m+1}$-CCC.
 \end{theorem}
 $Proof:$ By Theorem \ref{old NGBTD}, \ref{NGBTD(3,m)} and \ref{NGBTD(5,m)}, there exists an NGBTD$(k,m)$ for
 any integers $m$ and $k$ where $m\geq 1$, $2 \leq k\leq 5$ and $(k,m)\not\in (3,3)$.
 Then the conclusion follows from Theorem \ref{CCC+GDRP}. \qed


 \vspace{0.3cm} \noindent{\bf Acknowledgement}

The authors thank the referees for their valuable comments and
suggestions.

This work is supported by NSFC under Grants No. 10801064, 11001109,
and 10926103, and by the Program for Innovative Research Team of
Jiangnan University.


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\end{document}
