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\newcommand{\zz}{\mathbb{Z}}

\title{\bf Enumeration of (16,4,16,4)\\ Relative Difference Sets }


\author{David Clark\\
\small Department of Mathematics\\[-0.8ex]
\small University of Minnesota\\[-0.8ex]
\small Minneapolis, MN 55455, USA\\
\small\tt dcclark@umn.edu\\
\and
Vladimir D. Tonchev\thanks{
The authors acknowledge support by  NSA Grant H98230-12-0213.}\\
\small Department of Mathematics\\[-0.8ex]
\small Michigan Technological University\\[-0.8ex]
\small Houghton, MI 49931, USA\\
\small\tt tonchev@mtu.edu}

\date{\dateline{Oct 5, 2012}{Mar 21, 2013}{Mar 31, 2013}
%\small Mathematics Subject Classifications: 05C88, 05C89
}

\begin{document}

\maketitle

\begin{abstract}
A complete enumeration of relative difference sets
(RDS) with parameters $(16,4,16,4)$ in a group of order 64 with a normal 
subgroup $N$ of order 4 is given.
If $N=Z_4$, three of the eleven abelian groups of order 64, and 23 of the 256 
nonabelian groups of order 64 contain  $(16,4,16,4)$ RDSs.
If $N=Z_2 \times Z_2$, six of the abelian groups  
and 194 of the non-abelian groups of order 64 contain  $(16,4,16,4)$  RDSs.

\bigskip\noindent\textbf{Keywords:} Relative difference set; 
symmetric net.
\end{abstract}

\section{Introduction}

A relative difference set (RDS) with parameters $(m,n,k,\lambda)$ in a finite 
group $G$ of order $mn$ relative to a normal subgroup $N$ of order $n$ 
is a $k$-subset $R$ of $G$ such that every element of $g\in G\setminus N$ appears 
exactly $\lambda$ times in the multiset $S=\{ ab^{-1} \ | \ a,b \in R, a\neq b \}$, 
and no element of $N$ appears in $S$ \cite{BJL}. 
An RDS is called {\em abelian} if $G$ is abelian, and {\it nonabelian} otherwise.

Relative difference sets are closely related
to difference sets, group-divisible designs, generalized Hadamard matrices,
symmetric nets, and finite geometry \cite{BJL}, \cite{HLT}, \cite{R}.  
A comprehensive survey on RDS is the paper by Pott \cite{Pott}. 
The existence problem of $(p^a,p^b,p^a,p^{a-b})$ RDSs is
considered to be one of the most important questions concerning RDSs \cite{Pott}. 

In \cite{S}, Schmidt studied the existence of abelian  $(p^a,p^b,p^a,p^{a-b})$ RDS,
and settled the existence problem of abelian $(16,4,16,4)$ RDS completely.

In this paper, a complete enumeration of  $(16,4,16,4)$ RDSs is given,
for all groups, abelian and nonabelian, of order 64. 
In summary, RDSs exist in 6 of the 11 abelian groups of order 64, as well as 
in 195 of the 256 nonabelian groups of order 64.
If $N=Z_4$, three of the eleven abelian groups of order 64, and 23 of the 256
non-abelian groups of order 64 contain $(16,4,16,4)$ RDSs.
If $N=Z_2 \times Z_2$, six of the abelian groups
and 194 of the non-abelian groups of order 64 contain $(16,4,16,4)$  RDSs.
The computer algebra package Magma \cite{Magma} was used in the computations.

\section{RDS and symmetric nets}


Our approach to the enumeration of $(16,4,16,4)$ RDSs is based on their link 
to incidence structures known as symmetric $(4,4)$-nets. 

A {\it symmetric $(4,4)$-net}\footnote{More generally, a
net is defined as a resolvable 1-design, and a symmetric net is 
a net with equal number of points and blocks.
For more definitions concerning  designs see \cite{BJL}.} 
is an incidence structure $I=(X,\cal{B})$
consisting of a set $X$ of 64 {\it points} and a collection $\cal{B}$ 
of 64 {\it blocks},
each block being a subset of 16 points of $X$, having the following
properties:
\begin{itemize}
\item Each point belongs to 16 blocks.
\item There exists a partition $\cal{P}$
of the point set $X$ into 16 subsets of size 4, called {\it groups}, 
so that every two points belonging to different groups
appear together in exactly 4 blocks, while every two points belonging
to the same group do not appear together in any block.
\item The 64 blocks are partitioned into 16 {\it parallel classes}, each class
consisting of 4 pairwise disjoint blocks, so that every two blocks belonging to
different parallel classes share exactly 4 points.
\end{itemize}
Other terms used for a structure with the above properties are group-divisible design,
or a transversal design \cite{BJL}.

An {\it automorphism} of an incidence structure $I$ is any permutation 
of the point set which preserves the collection of blocks. 
The set of all automorphisms of $I$ form a group, called the {\em full} 
automorphism group, $Aut(I)$, of $I$. The subgroups of $Aut(I)$ are
called automorphism groups. 

A symmetric $(4,4)$-net is  {\it class-regular} if it admits an
automorphism group $N$ of order 4 which acts transitively on each group
of points and each parallel class of blocks. The group $N$ is then called
a group of {\it bitranslations}. 

If $R$ is a $(16,4,16,4)$ RDS in a group $G$ of order 64, relative to a
normal subgroup
$N\leq G$ of order 4, one can associate with $R$ a class-regular $(4,4)$-net $I$
with point set $G$ and  blocks being the subsets  $B_g\subseteq G$ of the form 
\[ B_g=\{ Rg \ | \ g\in G \}. \]
The partition $\cal{P}$ of the points into subsets of size 4
is defined as the partition of $G$ into cosets of $N$.
Consequently, $G$ acts as an automorphism group of $I$, and the subgroup $N$
acts transitively on each point group and each parallel class.

Thus, any $(16,4,16,4)$ RDS corresponds to a class-regular
symmetric $(4,4)$-net which admits a regular automorphism group.

All nonisomorphic class-regular symmetric $(4,4)$-nets were enumerated 
by Harada, Lam and Tonchev in \cite{HLT},
and, implicitly, by Gibbons and Mathon in \cite{GM} 
(two incidence structures are isomorphic 
if there is an incidence preserving bijection between their point sets).
Up to isomorphism, there are exactly 226 nets with group of
 bitranslations $N=Z_2 \times Z_2$,
%available online at 
%\begin{verbatim}
%http://www.math.mtu.edu/~tonchev/Z2Z2nets
%\end{verbatim}
and 13 nets with $N=Z_4$.
% available at
%\begin{verbatim}
%http://www.math.mtu.edu/~tonchev/Z4nets
%\end{verbatim}

These results reduce the enumeration of  $(16,4,16,4)$ RDSs to finding
sharply transitive regular 
subgroups $G$ of the full automorphism groups of those class-regular symmetric 
$(4,4)$-nets which admit automorphism groups acting transitively on the points, 
such that $N$ is a normal subgroup of $G$.
We used Magma to find the conjugacy classes of sharply transitive regular subgroups. 

There are 267 groups of order 64, of which 11 are abelian and 256 are nonabelian.
 Among the 226 nets with group of bitranslations $Z_{2} \times Z_{2}$, 
only 200 nonisomorphic regular subgroups of order 64 appeared 
within the automorphism groups of those nets, of which 6 were abelian and 194 
were nonabelian. Among the 13 nets with group of bitranslations $Z_{4}$, 
only 26 nonisomorphic regular subgroups of order 64 appeared within 
the automorphism groups of those nets, of which 3 were abelian and 23 were nonabelian.

\section{The results}


Tables \ref{z2z2table} and \ref{z4table} list the nets
with automorphism groups which admit regular subgroups
with normal subgroup $N$.
Each entry is as follows:
\begin{itemize}
\item \#: The index of the net within the list of nets with a group of bitranslations $N=\zz_{2} \times \zz_{2}$ available at \url{http://www.math.mtu.edu/~tonchev/Z2Z2nets}\\ 
and at \url{http://www.math.mtu.edu/~tonchev/Z4nets} for the nets with  $N=\zz_{4}$. Missing indices indicate that the 
corresponding nets do not have transitive automorphism groups.
\item \textit{Order:} The order of the automorphism group.
\item \textit{2-Rank:} The 2-rank of the incidence matrix of the net.
\item \textit{Total:} In the format $x/y$, $x$ indicates the total number 
of conjugacy classes of regular subgroups containing the group of bitranslations, 
found within the automorphism group of each net, while $y$ is the number of 
nonisomorphic regular subgroups.
\item \textit{Abelian} and \textit{Nonabelian:} In the format $x/y$,
 $x$ indicates the number of conjugacy classes of each type of subgroup 
found within the net's automorphism group, while $y$ is the number of nonisomorphic
 subgroups.
\item \textit{List of indices:} Below the previous data is a list of the 
indices of the regular subgroups of order 64 found in each automorphism group 
according to Magma's list of the groups of order 64. 
Entries of the form $x (y)$ indicate that groups isomorphic to group $x$ 
appeared in $y$ distinct conjugacy classes. Entries marked with an 
asterisk are abelian, and all others are nonabelian.
\end{itemize}

Tables \ref{z2z2structure} and \ref{z4structure} give details about the 
structure of the regular abelian subgroups of order 64 found. 
Only such subgroups containing the relevant group $N$ of bitranslations 
were considered.
 Groups with the following indices had a single regular abelian subgroup isomorphic
 to $\zz_{2}^{6}$, and are not listed in Table \ref{z2z2structure}: 
 6, 99, 100, 103, 104, 105, 107, 111, 113, 120, 121, 127, 128, 131, 142, 143, 
144, 145, 146, 147, 148, 149, 150, 151, 181, 182, 192, 193, 194, 195, 196, 
197, 198, 212, 214, 221, 224, 225, 226.
% Again, a missing index indicates that the associated net 
%had no such regular subgroups. We give a presentation as product of cyclic 
%groups for each nonisomorphic subgroup found.

Tables \ref{allz4Abelian} and \ref{allz2z2Abelian} summarize the structures of 
the regular abelian subgroups of order 64 found within the nets. 

Table \ref{notAppearingAbelian} gives the structures of all abelian groups of 
order 64 which do not appear as a regular subgroup  
in any net. Consequently, those are the groups which do not support any
$(16,4,16,4)$ RDS.

%Finally, Table \ref{nonAbelianOrderTable} lists the exponent of each 
%abelian regular subgroup. This refers to the exponent bounds found 
%in \cite{bschm}, which establish conditions for existence of 
%relative difference sets.
		
	\begin{longtable}{p{0.25in}*{5}{p{0.75in}}}
	\caption{$(16,4,16,4)$ RDS with $N=\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ } \\
	\label{z2z2table}
		\# & Order & 2-Rank & Total & Abelian & Nonabelian \\
		\hline\endhead
		
		%%%% START Z2xZ2 nets %%%%
		
1 & 1105920 & 16 & \emph{173/57} & \emph{6/2} &\emph{167/55} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4(2), 9(2), 18(2), 20(2), 23(5), 25(2), 32(6), 33(2), 34, 35(3), 36, 37(3), 56(8), 60(2), 74, 77, 80, 88(4), 90(5), 92(2), 100(2), 102(2), 132, 165, 192(4)*, 193(4), 194, 198(2), 199(3), 200(2), 202(3), 206(3), 207(3), 210(4), 214(3), 215(3), 217(3), 219(4), 223(8), 224(5), 226(2), 227(4), 229(2), 230(2), 232(10), 236, 237(6), 239(4), 240, 241(2), 242(9), 243, 244(5), 245, 262(3), 264(2), 267(2)*}} \\[0.5ex]
\hline
2 & 13824 & 18 & \emph{8/7} & \emph{2/2} &\emph{6/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(2), 88, 132, 192*, 193, 262, 267*}} \\[0.5ex]
\hline
4 & 4608 & 18 & \emph{8/5} & \emph{0/0} &\emph{8/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{56(2), 88(2), 100, 132, 262(2)}} \\[0.5ex]
\hline
5 & 18432 & 16 & \emph{61/21} & \emph{6/2} &\emph{55/19} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4(2), 9(2), 20, 23, 32(4), 35(2), 56(14), 60(2), 74(3), 88(3), 90, 92, 100, 132(2), 192(5)*, 193(5), 202, 207(2), 242(5), 262(3), 267*}} \\[0.5ex]
\hline
6 & 4608 & 20 & \emph{28/19} & \emph{1/1} &\emph{27/18} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 58, 60(6), 69(2), 71, 74, 75, 78, 90, 92, 100, 109, 193, 202(2), 204, 207, 247, 267*}} \\[0.5ex]
\hline
8 & 384 & 20 & \emph{4/3} & \emph{0/0} &\emph{4/3} \\
& \multicolumn{5}{l}{\parbox{4.5in}{88, 132(2), 262}} \\[0.5ex]
\hline
9 & 4608 & 18 & \emph{8/7} & \emph{2/2} &\emph{6/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(2), 88, 132, 192*, 193, 262, 267*}} \\[0.5ex]
\hline
\pagebreak
22 & 6144 & 16 & \emph{84/25} & \emph{6/2} &\emph{78/23} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4(2), 9(2), 20, 23, 32(2), 56(29), 60, 66, 67(2), 69(3), 71(2), 73(2), 75(3), 78(2), 88(3), 90, 92, 109, 131, 164, 192(5)*, 193(8), 194(8), 260*, 261}} \\[0.5ex]
\hline
24 & 1536 & 20 & \emph{48/35} & \emph{0/0} &\emph{48/35} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18, 22, 25(2), 88(2), 93(2), 97, 100(5), 102, 111, 114, 116, 117, 120, 122, 132(5), 133, 143, 145, 149, 151, 156, 158, 163, 166(3), 168, 170, 200, 206, 217, 229, 230, 249, 252, 255, 262}} \\[0.5ex]
\hline
25 & 1536 & 18 & \emph{9/5} & \emph{0/0} &\emph{9/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{56(3), 88(2), 100, 132, 262(2)}} \\[0.5ex]
\hline
30 & 1536 & 20 & \emph{50/42} & \emph{0/0} &\emph{50/42} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18, 22, 25(2), 33, 36, 91, 93(2), 97(2), 99, 100(2), 105, 111, 119, 120, 121, 122, 129, 130, 131, 132(2), 133, 142, 143, 144, 145(3), 146, 148, 149, 151, 156, 158, 166(2), 168, 170, 176, 178, 200, 249, 251, 252, 254, 255}} \\[0.5ex]
\hline
31 & 1536 & 20 & \emph{45/26} & \emph{1/1} &\emph{44/25} \\
& \multicolumn{5}{l}{\parbox{4.5in}{23, 24, 32(6), 35(4), 58, 59, 60, 61(2), 66, 67(2), 69, 72(2), 74, 75, 78, 85, 90(2), 101(2), 136(3), 138(2), 202(2), 207, 212, 242(2), 255(2), 260*}} \\[0.5ex]
\hline
32 & 128 & 20 & \emph{4/4} & \emph{1/1} &\emph{3/3} \\
& \multicolumn{5}{l}{\parbox{4.5in}{146, 148, 206, 246*}} \\[0.5ex]
\hline
33 & 512 & 20 & \emph{66/44} & \emph{1/1} &\emph{65/43} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18, 25, 32(4), 33(4), 35(2), 86(2), 88, 91(2), 97(2), 99, 100, 102(2), 105, 108(3), 115, 116, 119, 122, 132, 133, 135, 138, 139(3), 145, 146(2), 148(2), 149, 151, 155, 158(2), 163, 166(3), 170, 179, 206(2), 217, 223(2), 227, 232, 237, 242, 246*, 251, 254}} \\[0.5ex]
\hline
36 & 73728 & 16 & \emph{1158/85} & \emph{6/3} &\emph{1152/82} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4(2), 9(2), 18(6), 20(6), 23(13), 25(6), 32(14), 33(10), 34(5), 35(7), 36(5), 37(7), 56(13), 60(3), 62, 74(4), 77(2), 80, 88(6), 90(11), 92(3), 93, 99(2), 100(4), 102(6), 132(2), 192(4)*, 193(6), 194(2), 195(5), 196(5), 197(3), 198(13), 199(15), 200(5), 201(6), 202(7), 203(6), 204(6), 205(7), 206(29), 207(7), 209(8), 210(52), 211(3), 212(3), 213(12), 214(17), 215(16), 216(12), 217(19), 218(9), 219(56), 220(44), 221(18), 222(26), 223(68), 224(12), 225(13), 226(15), 227(58), 228(19), 229(22), 230(11), 231(5), 232(69), 233(46), 234(38), 235(23), 236(20), 237(33), 238(9), 239(5), 240(18), 241(27), 242(15), 243(24), 244(31), 260*, 261, 262(4), 263, 264(5), 265, 267*}} \\[0.5ex]
\hline
78 & 6144 & 20 & \emph{28/24} & \emph{0/0} &\emph{28/24} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4, 5, 9(4), 57, 62, 67, 68, 69(2), 70, 73, 74, 77, 78, 79, 81, 82, 87, 112, 131, 132, 164, 165, 208, 263}} \\[0.5ex]
\hline
99 & 1152 & 20 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
100 & 1152 & 22 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
103 & 384 & 20 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
104 & 384 & 22 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
105 & 1152 & 18 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
107 & 6912 & 20 & \emph{5/3} & \emph{1/1} &\emph{4/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 193, 267*}} \\[0.5ex]
\hline
109 & 384 & 22 & \emph{4/4} & \emph{0/0} &\emph{4/4} \\
& \multicolumn{5}{l}{\parbox{4.5in}{100, 109, 204, 247}} \\[0.5ex]
\hline
111 & 512 & 20 & \emph{52/19} & \emph{1/1} &\emph{51/18} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 58(2), 60(10), 69(6), 71(2), 74(3), 75(3), 78(3), 90, 92(2), 100, 109, 193(2), 202(6), 204(2), 207(2), 247, 267*}} \\[0.5ex]
\hline
113 & 1536 & 20 & \emph{48/19} & \emph{1/1} &\emph{47/18} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 58(2), 60(10), 69(6), 71(2), 74(3), 75(3), 78(3), 90, 92, 100, 109, 193(2), 202(3), 204(2), 207(2), 247, 267*}} \\[0.5ex]
\hline
115 & 256 & 20 & \emph{4/2} & \emph{0/0} &\emph{4/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{9(2), 132(2)}} \\[0.5ex]
\hline
 116 & 768 & 20 & \emph{4/2} & \emph{0/0} &\emph{4/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{9(2), 132(2)}} \\[0.5ex]
\hline
117 & 128 & 21 & \emph{0/0} & \emph{0/0} &\emph{0/0} \\
& \multicolumn{5}{l}{\parbox{4.5in}{}} \\[0.5ex]
\hline
120 & 1536 & 20 & \emph{31/19} & \emph{1/1} &\emph{30/18} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 58, 60(6), 69(2), 71, 74, 75, 78, 90, 92(2), 100, 109, 193, 202(4), 204, 207, 247, 267*}} \\[0.5ex]
\hline
121 & 384 & 22 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
122 & 128 & 22 & \emph{4/4} & \emph{0/0} &\emph{4/4} \\
& \multicolumn{5}{l}{\parbox{4.5in}{100, 109, 204, 247}} \\[0.5ex]
\hline
123 & 192 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20}} \\[0.5ex]
\hline
124 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{23}} \\[0.5ex]
\hline
\pagebreak
125 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{132}} \\[0.5ex]
\hline
126 & 192 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{132}} \\[0.5ex]
\hline
127 & 768 & 20 & \emph{6/3} & \emph{1/1} &\emph{5/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(4), 193, 267*}} \\[0.5ex]
\hline
128 & 256 & 20 & \emph{7/3} & \emph{1/1} &\emph{6/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(4), 193(2), 267*}} \\[0.5ex]
\hline
131 & 768 & 20 & \emph{6/3} & \emph{1/1} &\emph{5/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(4), 193, 267*}} \\[0.5ex]
\hline
132 & 512 & 20 & \emph{52/32} & \emph{1/1} &\emph{51/31} \\
& \multicolumn{5}{l}{\parbox{4.5in}{23, 24, 32, 33(2), 58, 59(2), 61, 62, 63, 65(2), 66, 67(3), 68(3), 69(7), 70, 71, 72, 75, 77(3), 78(2), 80, 81(2), 85, 90(2), 98(2), 101(2), 139, 192*, 197, 204, 205, 212}} \\[0.5ex]
\hline
134 & 256 & 20 & \emph{64/21} & \emph{0/0} &\emph{64/21} \\
& \multicolumn{5}{l}{\parbox{4.5in}{58, 59(2), 61, 63, 65, 68(2), 69(10), 70(5), 71(4), 72, 73(2), 75(9), 77(4), 78(7), 79(3), 80(3), 81(2), 195(2), 196, 197, 204(2)}} \\[0.5ex]
\hline
136 & 512 & 20 & \emph{52/28} & \emph{1/1} &\emph{51/27} \\
& \multicolumn{5}{l}{\parbox{4.5in}{23, 24, 32, 35(2), 55*, 58(2), 59, 61, 63(3), 66(2), 67(4), 69(8), 71, 72(4), 74(3), 75, 76, 78(2), 81(2), 90(2), 98, 101(2), 104, 136, 197, 208, 255, 262}} \\[0.5ex]
\hline
137 & 768 & 20 & \emph{24/13} & \emph{0/0} &\emph{24/13} \\
& \multicolumn{5}{l}{\parbox{4.5in}{57, 59, 69(4), 70(3), 72, 77(3), 78(2), 79(2), 80(3), 81, 82, 197, 212}} \\[0.5ex]
\hline
 138 & 1536 & 20 & \emph{45/30} & \emph{1/1} &\emph{44/29} \\
& \multicolumn{5}{l}{\parbox{4.5in}{23, 24, 32(2), 33(4), 34(4), 55*, 58, 61, 62(2), 65, 67, 69, 70, 71, 75, 77, 79, 90(2), 98, 101(2), 104, 134(2), 139(3), 194, 196, 203, 205(2), 209, 241(2), 254}} \\[0.5ex]
\hline
139 & 512 & 21 & \emph{52/40} & \emph{0/0} &\emph{52/40} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18, 20, 25(2), 89, 91(2), 92, 95, 97, 98, 100(3), 102(2), 105, 112, 115, 117, 119, 120, 129, 131(2), 132, 133(2), 145(2), 148, 151, 159, 160, 163, 164, 165, 166, 169(2), 198, 200, 207(2), 217(2), 227(2), 229, 252, 255, 264}} \\[0.5ex]
\hline
140 & 512 & 21 & \emph{66/44} & \emph{0/0} &\emph{66/44} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18, 25, 32(2), 33(6), 35(2), 86(2), 88, 91(2), 97, 100(3), 102(2), 105, 108(2), 109, 116, 117, 120, 122, 132(2), 137, 138, 139(3), 143, 148(2), 151(4), 156, 158, 160, 165, 166(3), 168, 182, 204, 206, 217, 222, 225, 230, 232, 233, 244, 247, 252, 255}} \\[0.5ex]
\hline
\pagebreak
142 & 512 & 19 & \emph{52/19} & \emph{1/1} &\emph{51/18} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 32(2), 60(10), 66(2), 67(2), 69(4), 71(3), 77(4), 78(4), 88, 90, 92, 109(2), 193(2), 195(2), 202(5), 205(2), 267*}} \\[0.5ex]
\hline
143 & 6144 & 20 & \emph{45/33} & \emph{1/1} &\emph{44/32} \\
& \multicolumn{5}{l}{\parbox{4.5in}{5, 9(2), 20, 23(2), 32, 57, 58, 60(5), 61, 62, 66(2), 67, 68, 69(2), 70, 71, 77(2), 78, 79, 81, 88, 90(4), 92, 109, 112, 164, 165, 193, 195, 202, 205, 208, 267*}} \\[0.5ex]
\hline
144 & 128 & 20 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
145 & 256 & 20 & \emph{64/6} & \emph{1/1} &\emph{63/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(21), 69(21), 77(7), 80(7), 209(7), 267*}} \\[0.5ex]
\hline
146 & 64 & 22 & \emph{1/1} & \emph{1/1} &\emph{0/0} \\
& \multicolumn{5}{l}{\parbox{4.5in}{267*}} \\[0.5ex]
\hline
147 & 128 & 20 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
148 & 128 & 22 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
149 & 128 & 22 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
150 & 384 & 20 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
151 & 1152 & 21 & \emph{2/2} & \emph{1/1} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60, 267*}} \\[0.5ex]
\hline
152 & 512 & 21 & \emph{66/43} & \emph{0/0} &\emph{66/43} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18(2), 25(2), 32(2), 33(2), 34(3), 35(3), 91(6), 97(2), 99, 100(3), 105(2), 115(2), 117(2), 118, 119(2), 120, 129, 130, 132, 133(3), 144, 145, 146, 148, 149, 151, 159, 160, 164, 165, 200(2), 227, 229, 232, 233, 234, 236, 241, 244, 251, 252, 254, 255}} \\[0.5ex]
\hline
 156 & 256 & 25 & \emph{8/7} & \emph{0/0} &\emph{8/7} \\
& \multicolumn{5}{l}{\parbox{4.5in}{35, 84, 91, 97, 170, 172(2), 214}} \\[0.5ex]
\hline
 157 & 256 & 21 & \emph{68/48} & \emph{0/0} &\emph{68/48} \\
& \multicolumn{5}{l}{\parbox{4.5in}{86, 89, 95, 97, 98, 99, 105, 109(3), 116(2), 117(2), 118, 119, 122, 123, 128, 130, 133(2), 141, 144, 146(4), 147(2), 149(3), 150(2), 151, 155, 157(2), 159, 160(2), 162, 163(4), 164(2), 166, 170, 171, 179, 180, 216, 218, 219, 221(2), 223, 226, 227(2), 228, 232, 233, 248, 254}} \\[0.5ex]
\hline
\pagebreak
158 & 256 & 21 & \emph{68/47} & \emph{0/0} &\emph{68/47} \\
& \multicolumn{5}{l}{\parbox{4.5in}{85, 88, 96(2), 97, 100, 104, 106, 108, 109, 115(2), 116(2), 120(2), 121, 122, 129, 132(2), 133, 143, 145, 148(2), 149(3), 151(7), 158, 159, 160(4), 162, 163, 165(2), 166(4), 169, 172, 180, 182, 201, 204, 206, 210, 214, 220, 228, 229, 230, 232, 233, 235, 247, 252}} \\[0.5ex]
\hline
166 & 1024 & 20 & \emph{52/33} & \emph{1/1} &\emph{51/32} \\
& \multicolumn{5}{l}{\parbox{4.5in}{9(2), 20(2), 23(2), 33, 35, 58, 62(3), 63, 65, 66, 67(3), 68(2), 69(2), 72(3), 74, 75(2), 77, 85, 88(2), 90(2), 92(2), 93(4), 100, 104, 109, 114, 192*, 195(2), 204, 205, 212, 217, 262}} \\[0.5ex]
\hline
167 & 512 & 21 & \emph{52/40} & \emph{1/1} &\emph{51/39} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18(3), 20(2), 22, 25(2), 33, 36, 83*, 85, 92, 100, 105(3), 111, 112, 114, 119, 120, 121, 122, 131, 132, 145(2), 146, 148, 149, 151, 155, 156(2), 157, 158(2), 166(2), 169(2), 180(2), 199, 200, 207, 210, 214, 223, 232, 237}} \\[0.5ex]
\hline
168 & 512 & 19 & \emph{66/33} & \emph{0/0} &\emph{66/33} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 32(6), 33(2), 35(2), 58, 59, 62(2), 66(5), 67(4), 68(3), 69(2), 70(2), 71, 72, 75(3), 76, 78(2), 79(2), 90, 91(2), 93, 94(2), 100, 109, 194, 195(2), 196, 204(2), 205(2), 212, 232(4), 247}} \\[0.5ex]
\hline
169 & 512 & 20 & \emph{48/21} & \emph{0/0} &\emph{48/21} \\
& \multicolumn{5}{l}{\parbox{4.5in}{5(2), 9(4), 59(5), 63, 68(2), 69(8), 70, 72(2), 74, 76, 77(2), 78(4), 79(4), 80, 81, 113(2), 132, 164(2), 165, 197(2), 212}} \\[0.5ex]
\hline
175 & 512 & 20 & \emph{2/2} & \emph{0/0} &\emph{2/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4, 87}} \\[0.5ex]
\hline
177 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{132}} \\[0.5ex]
\hline
178 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{233}} \\[0.5ex]
\hline
181 & 384 & 21 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
182 & 1536 & 19 & \emph{76/29} & \emph{1/1} &\emph{75/28} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 32(4), 33(4), 34(2), 35(2), 60(3), 88, 90, 92, 99, 100, 193(2), 195, 196, 202(7), 203, 204, 205, 207, 209(2), 211, 216(4), 219(8), 227(8), 232(6), 241(6), 263, 267*}} \\[0.5ex]
\hline
 183 & 512 & 21 & \emph{52/39} & \emph{0/0} &\emph{52/39} \\
& \multicolumn{5}{l}{\parbox{4.5in}{25(2), 86(2), 91(2), 99, 100(3), 102(2), 108(2), 119, 120, 121, 122, 129, 131, 132(2), 135, 137, 139(2), 142, 143, 145(2), 146, 148, 149, 151, 156, 158, 159, 160, 163, 166(3), 168, 169, 178, 182, 217(2), 251, 252, 254, 255}} \\[0.5ex]
\hline
\pagebreak
 184 & 256 & 21 & \emph{64/38} & \emph{0/0} &\emph{64/38} \\
& \multicolumn{5}{l}{\parbox{4.5in}{96(2), 98, 99, 106, 109, 119(2), 120(2), 121(2), 122, 123, 129(3), 131(2), 132, 133(2), 142, 143, 144(2), 145(4), 146(2), 147(2), 151(4), 157, 158, 159(2), 160(4), 161(2), 162, 165(3), 166(2), 169(2), 170, 172, 176, 178, 180, 182, 252, 254}} \\[0.5ex]
\hline
185 & 512 & 19 & \emph{52/32} & \emph{0/0} &\emph{52/32} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20(2), 23(2), 33, 35, 58, 59, 62(2), 66(3), 67(3), 68(3), 69(4), 70(2), 71(2), 72, 74, 75(5), 76, 77(2), 79(2), 80, 90, 91, 93, 94, 100, 109, 194, 195, 204, 205, 212, 247}} \\[0.5ex]
\hline
186 & 512 & 21 & \emph{52/31} & \emph{1/1} &\emph{51/30} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18, 19, 83*, 85, 86(2), 88(2), 89(3), 91, 94, 97, 101, 102, 104(2), 105(2), 108, 110, 112(3), 114(3), 115(2), 116(4), 117(6), 126(2), 127(2), 135, 137, 138, 139, 206, 247, 248, 256}} \\[0.5ex]
\hline
187 & 512 & 20 & \emph{48/23} & \emph{0/0} &\emph{48/23} \\
& \multicolumn{5}{l}{\parbox{4.5in}{5(2), 9(4), 58, 59, 61, 62(2), 63, 66(4), 67(2), 69(2), 70(3), 72(2), 73, 75(5), 77, 78(3), 79(4), 113(2), 131, 164(3), 196, 204, 212}} \\[0.5ex]
\hline
190 & 1536 & 20 & \emph{28/11} & \emph{0/0} &\emph{28/11} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4(2), 9(4), 62(3), 67(4), 73(2), 74(3), 82(2), 87(2), 131(2), 132(2), 263(2)}} \\[0.5ex]
\hline
192 & 128 & 21 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
193 & 128 & 22 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
194 & 384 & 20 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
195 & 128 & 21 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
196 & 640 & 22 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
197 & 384 & 22 & \emph{2/2} & \emph{1/1} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60, 267*}} \\[0.5ex]
\hline
198 & 384 & 20 & \emph{4/2} & \emph{1/1} &\emph{3/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(3), 267*}} \\[0.5ex]
\hline
201 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20}} \\[0.5ex]
\hline
 202 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{23}} \\[0.5ex]
\hline
203 & 128 & 22 & \emph{4/4} & \emph{0/0} &\emph{4/4} \\
& \multicolumn{5}{l}{\parbox{4.5in}{97, 108, 206, 247}} \\[0.5ex]
\hline
204 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20}} \\[0.5ex]
\hline
 206 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{9}} \\[0.5ex]
\hline
207 & 128 & 22 & \emph{2/1} & \emph{0/0} &\emph{2/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{90(2)}} \\[0.5ex]
\hline

208 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{109}} \\[0.5ex]
\hline
209 & 64 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{233}} \\[0.5ex]
\hline
211 & 30720 & 18 & \emph{2316/78} & \emph{2/2} &\emph{2314/76} \\
& \multicolumn{5}{l}{\parbox{4.5in}{18(4), 20(4), 23(4), 25(4), 32(8), 33(8), 34(4), 35(4), 36(4), 37(4), 60, 62, 88(2), 90(2), 92, 93, 99(2), 100(2), 102(4), 192*, 193(2), 194, 195(9), 196(17), 197(5), 198(22), 199(17), 200(7), 201(22), 202(7), 203(17), 204(15), 205(17), 206(62), 207(7), 208(9), 209(10), 210(128), 211(2), 212(4), 213(32), 214(34), 215(29), 216(32), 217(36), 218(31), 219(132), 220(124), 221(66), 222(66), 223(116), 224(19), 225(31), 226(34), 227(122), 228(65), 229(46), 230(19), 231(19), 232(135), 233(122), 234(122), 235(65), 236(34), 237(65), 238(19), 239(11), 240(34), 241(47), 242(15), 243(63), 244(66), 245(5), 262(2), 263(4), 264(6), 265(2), 267*}} \\[0.5ex]
\hline
212 & 2048 & 20 & \emph{92/38} & \emph{1/1} &\emph{91/37} \\
& \multicolumn{5}{l}{\parbox{4.5in}{5, 9(2), 20, 23(4), 32, 58(3), 59, 60(10), 61(3), 62(3), 64, 66(6), 67(3), 68(5), 69(6), 70, 71(2), 74, 76, 77(4), 78(3), 79(2), 80, 81, 88, 90(8), 92, 109, 112, 132, 164, 193(2), 195(2), 197, 202(2), 205(3), 208, 267*}} \\[0.5ex]
\hline
213 & 2048 & 20 & \emph{48/27} & \emph{0/0} &\emph{48/27} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4, 5, 9(4), 59, 62(2), 64, 67(3), 68(5), 69(6), 70, 73, 74(3), 76, 77, 78(3), 79(2), 80, 81, 82, 87, 112, 131, 132(2), 164, 197, 208, 263}} \\[0.5ex]
\hline
214 & 1024 & 20 & \emph{32/23} & \emph{1/1} &\emph{31/22} \\
& \multicolumn{5}{l}{\parbox{4.5in}{4, 9(2), 24(2), 55*, 57, 59, 66(2), 68, 69(2), 71(3), 72, 75(2), 77, 80, 87, 88, 89, 197, 207, 208(2), 212(2), 222, 247}} \\[0.5ex]
\hline
221 & 768 & 19 & \emph{24/6} & \emph{1/1} &\emph{23/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(7), 69(7), 77(3), 80(3), 209(3), 267*}} \\[0.5ex]
\hline
222 & 256 & 22 & \emph{8/8} & \emph{0/0} &\emph{8/8} \\
& \multicolumn{5}{l}{\parbox{4.5in}{35, 90, 114, 131, 145, 169, 178, 222}} \\[0.5ex]
\hline
\pagebreak
223 & 192 & 22 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{20}} \\[0.5ex]
\hline
224 & 768 & 20 & \emph{24/6} & \emph{1/1} &\emph{23/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(7), 69(7), 77(3), 80(3), 209(3), 267*}} \\
\hline
225 & 256 & 20 & \emph{64/6} & \emph{1/1} &\emph{63/5} \\
& \multicolumn{5}{l}{\parbox{4.5in}{60(21), 69(21), 77(7), 80(7), 209(7), 267*}} \\
\hline
226 & 320 & 21 & \emph{1/1} & \emph{1/1} &\emph{0/0} \\
& \multicolumn{5}{l}{\parbox{4.5in}{267*}}\\
\hline

	%%%% END Z2 x Z2 NETS %%%%
		
	\end{longtable}
	
	%\newpage
	
	\begin{longtable}{p{0.25in}*{5}{p{0.75in}}}
		\caption{$(16,4,16,4)$ RDS with $N=\mathbb{Z}_{4}$ } \\
		\label{z4table}
		\# & Order & 2-Rank & Total & Abelian & Nonabelian \\
		\hline
		
		%%%% START Z4 NETS %%%%
		
1 & 512 & 22 & \emph{14/9} & \emph{0/0} &\emph{14/9} \\
& \multicolumn{5}{l}{\parbox{4.5in}{58, 59(2), 61(2), 72(4), 85, 175, 208, 212, 262}} \\[0.5ex]
\hline
5 & 256 & 22 & \emph{16/10} & \emph{0/0} &\emph{16/10} \\
& \multicolumn{5}{l}{\parbox{4.5in}{59, 61, 63, 65, 70(4), 72(2), 195, 197(2), 204(2), 212}} \\[0.5ex]
\hline
6 & 512 & 22 & \emph{14/13} & \emph{0/0} &\emph{14/13} \\
& \multicolumn{5}{l}{\parbox{4.5in}{58, 59, 61, 63, 65, 70(2), 72, 85, 168, 194, 197, 204, 212}} \\[0.5ex]
\hline
8 & 512 & 22 & \emph{14/10} & \emph{2/2} &\emph{12/8} \\
& \multicolumn{5}{l}{\parbox{4.5in}{55*, 58(2), 59(3), 61, 70, 104, 142, 192*, 195, 196(2)}} \\[0.5ex]
\hline
9 & 7680 & 22 & \emph{9/8} & \emph{2/2} &\emph{7/6} \\
& \multicolumn{5}{l}{\parbox{4.5in}{55*, 57, 58(2), 61, 104, 143, 197, 260*}} \\[0.5ex]
\hline
10 & 128 & 22 & \emph{4/4} & \emph{0/0} &\emph{4/4} \\
& \multicolumn{5}{l}{\parbox{4.5in}{59, 72, 197, 212}} \\[0.5ex]
\hline
11 & 384 & 22 & \emph{2/2} & \emph{1/1} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{55*, 70}} \\[0.5ex]
\hline
12 & 64 & 24 & \emph{1/1} & \emph{0/0} &\emph{1/1} \\
& \multicolumn{5}{l}{\parbox{4.5in}{238}} \\[0.5ex]
\hline
13 & 384 & 22 & \emph{2/2} & \emph{0/0} &\emph{2/2} \\
& \multicolumn{5}{l}{\parbox{4.5in}{57, 212}} \\[0.5ex]
\hline


	%%%% END Z4 NETS %%%%
		
	\end{longtable}

\newpage
	
	\begin{longtable}{|r|l|l|l|l|}
	\caption{Structures of abelian regular subgroups in 
$\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ nets}\\
		\label{z2z2structure}
		\# & Order & Abelian & Group structure \\
		\hline
		
		%%%% START Z2xZ2 ABELIAN PRODUCTS %%%%
		
1 & 1105920 & \emph{6/2} &$\zz_2^2 \times \zz_4^2(4), \zz_2^6(2)$\\\hline
2 & 13824 & \emph{2/2} &$\zz_2^2 \times \zz_4^2, \zz_2^6$\\\hline
5 & 18432 & \emph{6/2} &$\zz_2^2 \times \zz_4^2(5), \zz_2^6$\\\hline
%6 & 4608 & \emph{1/1} &$\zz_2^6$\\\hline
9 & 4608 & \emph{2/2} &$\zz_2^2 \times \zz_4^2, \zz_2^6$\\\hline
22 & 6144 & \emph{6/2} &$\zz_2^2 \times \zz_4^2(5), \zz_2^4 \times \zz_4$\\\hline
31 & 1536 & \emph{1/1} &$\zz_2^4 \times \zz_4$\\\hline
32 & 128 & \emph{1/1} &$\zz_2^3 \times \zz_8$\\\hline
33 & 512 & \emph{1/1} &$\zz_2^3 \times \zz_8$\\\hline
36 & 73728 & \emph{6/3} &$\zz_2^2 \times \zz_4^2(4), \zz_2^4 \times \zz_4, \zz_2^6$\\\hline
%99 & 1152 & \emph{1/1} &$\zz_2^6$\\\hline
%100 & 1152 & \emph{1/1} &$\zz_2^6$\\\hline
%103 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
%104 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
%105 & 1152 & \emph{1/1} &$\zz_2^6$\\\hline
%107 & 6912 & \emph{1/1} &$\zz_2^6$\\\hline
%111 & 512 & \emph{1/1} &$\zz_2^6$\\\hline
%113 & 1536 & \emph{1/1} &$\zz_2^6$\\\hline
%120 & 1536 & \emph{1/1} &$\zz_2^6$\\\hline
%121 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
%127 & 768 & \emph{1/1} &$\zz_2^6$\\\hline
%128 & 256 & \emph{1/1} &$\zz_2^6$\\\hline
%131 & 768 & \emph{1/1} &$\zz_2^6$\\\hline
132 & 512 & \emph{1/1} &$\zz_2^2 \times \zz_4^2$\\\hline
136 & 512 & \emph{1/1} &$\zz_4^3$\\\hline
138 & 1536 & \emph{1/1} &$\zz_4^3$\\\hline
%142 & 512 & \emph{1/1} &$\zz_2^6$\\\hline
%143 & 6144 & \emph{1/1} &$\zz_2^6$\\\hline
%144 & 128 & \emph{1/1} &$\zz_2^6$\\\hline
%145 & 256 & \emph{1/1} &$\zz_2^6$\\\hline
%146 & 64 & \emph{1/1} &$\zz_2^6$\\\hline
%147 & 128 & \emph{1/1} &$\zz_2^6$\\\hline
%148 & 128 & \emph{1/1} &$\zz_2^6$\\\hline
%149 & 128 & \emph{1/1} &$\zz_2^6$\\\hline
%150 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
%151 & 1152 & \emph{1/1} &$\zz_2^6$\\\hline
166 & 1024 & \emph{1/1} &$\zz_2^2 \times \zz_4^2$\\\hline
167 & 512 & \emph{1/1} &$\zz_2 \times \zz_4 \times \zz_8$\\\hline
%181 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
%182 & 1536 & \emph{1/1} &$\zz_2^6$\\\hline
186 & 512 & \emph{1/1} &$\zz_2 \times \zz_4 \times \zz_8$\\\hline
%192 & 128 & \emph{1/1} &$\zz_2^6$\\\hline
%193 & 128 & \emph{1/1} &$\zz_2^6$\\\hline
%194 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
%195 & 128 & \emph{1/1} &$\zz_2^6$\\\hline
%196 & 640 & \emph{1/1} &$\zz_2^6$\\\hline
%197 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
%198 & 384 & \emph{1/1} &$\zz_2^6$\\\hline
211 & 30720 & \emph{2/2} &$\zz_2^2 \times \zz_4^2, \zz_2^6$\\\hline
%212 & 2048 & \emph{1/1} &$\zz_2^6$\\\hline
214 & 1024 & \emph{1/1} &$\zz_4^3$\\\hline
%221 & 768 & \emph{1/1} &$\zz_2^6$\\\hline
%224 & 768 & \emph{1/1} &$\zz_2^6$\\\hline
%225 & 256 & \emph{1/1} &$\zz_2^6$\\\hline
%226 & 320 & \emph{1/1} &$\zz_2^6$\\\hline

                
                %%%% END Z2 x Z2 ABELIAN PRODUCTS %%%%
                
	\end{longtable}

	\begin{longtable}{|r|l|l|l|}
		\caption{Structures of abelian regular subgroups  in $\mathbb{Z}_{4}$ nets.} \\
		\label{z4structure}
		\# & Order & Abelian & Group structure \\
		\hline
		
		%%%% START Z4 ABELIAN PRODUCTS %%%%
		
	8 & 512 &  \emph{2 / 2} & {{ $\zz_{4}^{3}, \zz_{2}^{2} \times \zz_{4}^{2}$ }} \\[0.5ex]

                \hline
9 & 7680 &  \emph{2 / 2} &
               {{ $\zz_{4}^{3}, \zz_{2}^{4} \times \zz_{4}$ }} \\[0.5ex]

                \hline
11 & 384 &  \emph{1 / 1} &
                {{ $\zz_{4}^{3}$ }} \\[0.5ex]
\hline

		
	\end{longtable}
	
	\begin{longtable}{ll}
	\caption{All regular abelian subgroups of order 64 appearing 
in GDDs with bitranslation group $\zz_{4}$.} \\
		\label{allz4Abelian}
		Group & Structure \\
		\hline
		55 & $\zz_{4}^{3}$ \\
		192 & $\zz_{2}^{2} \times \zz_{4}^{2}$ \\
		260 & $\zz_{2}^{4} \times \zz_{4}$
	\end{longtable}
	
		\begin{longtable}{ll}
\caption{All regular abelian subgroups of order 64 appearing in 
GDDs with bi-translation group $\zz_{2} \times \zz_{2}$.} \\
		\label{allz2z2Abelian}
		Group & Structure \\
		\hline
		55 & $\zz_{4}^{3}$ \\
		83 & $\zz_{2} \times \zz_{4} \times \zz_{8}$ \\
		192 & $\zz_{2}^{2} \times \zz_{4}^{2}$ \\
		246 & $\zz_{2}^{3} \times \zz_{8}$ \\
		260 & $\zz_{2}^{4} \times \zz_{4}$ \\
		267 & $\zz_{2}^{6}$
	\end{longtable}	

	
	\begin{longtable}{l|l}
\caption{Abelian groups which do not contain any $(16,4,16,4)$ RDS}\\ 
		\label{notAppearingAbelian}
		Group & Structure \\
		\hline
		1 & $\zz_{64}$ \\
		2 & $\zz_{8} \times \zz_{8}$ \\
		26 & $\zz_{4} \times \zz_{16}$ \\
		50 & $\zz_{2} \times \zz_{32}$ \\
		183 & $\zz_{2}^{2} \times \zz_{16}$ \\
	\end{longtable}
	

	
\subsection*{Acknowledgments}

The authors wish to thank the referees for their useful comments.
This research was partially supported by  NSA Grant H98230-12-0213.
	

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\end{document}
