Some new operations on $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices
%\boldmath Following the ideas of \cite{AGG11} about $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices, we introduce the notion of diagram, which visually represents any set of coboundaries. Diagrams are a very useful tool for the description and the study of paths and intersections, as described in \cite{AGG11}. Then, we will study four different operations on $\Z _t \times \Z_2^2$-cocyclic matrices. These operations will be defined on the set of coboundaries defining the matrix, preserve the Hadamard character of the cocyclic matrices, and allow us to obtain new Hadamard matrices from old ones. We split the set of Hadamard matrices into disjoint orbits, define representatives for them and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way.
1112.4300.tex
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\begin{document}

\title{Some new operations on $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices}

\author{V\'{\i}ctor~\'{A}lvarez\thanks{All authors are partially supported by FEDER funds via the research projects FQM-016 and P07--FQM--02980 from JJAA and MTM2008-06578 from MICINN (Spain).},~%~\IEEEmembership{Member,~IEEE,
%        Jos\'{e}~Andr\'{e}s~Armario,~%\IEEEmembership{Fellow,~OSA,
%        Mar\'{\i}a~Dolores~Frau,~%\IEEEmembership{Fellow,~OSA,
        F\'{e}lix~Gudiel\thanks{V. \'{A}lvarez and F. Gudiel are with the Department
of Applied Mathematics 1, University of Seville, Seville,
Spain. E-mail: \{valvarez,gudiel\}@us.es.},~%\IEEEmembership{Fellow,~OSA,
        and~Maria~Bel\'{e}n~G\"{u}emes\thanks{M.B. G\"{u}emes is with the Department
of Algebra, University of Seville, Seville,
Spain. E-mail: bguemes@us.es.}%,~%\IEEEmembership{Fellow,~OSA,
%        Elena~Mart\'{\i}n,~%\IEEEmembership{Fellow,~OSA,
%        and~Amparo~Osuna%,~\IEEEmembership{Life~Fellow,~IEEE}% <-this % stops a space
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%\thanks{Manuscript received November 7, 2011}
}%; revised January 11, 2007.


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%{On $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices}
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\begin{abstract}
%\boldmath
Following the ideas of \cite{AGG11} about $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices,  we introduce the notion of diagram, which
visually represents any set of coboundaries. Diagrams are a very
useful tool for the description and the study of paths and intersections, as described in \cite{AGG11}. Then, we will study four different operations on
$\Z _t \times \Z_2^2$-cocyclic matrices. These operations will be defined on the set of coboundaries defining the matrix, preserve the Hadamard character of the cocyclic matrices,
and allow us to obtain new Hadamard matrices from old ones. We split the set of Hadamard matrices into disjoint orbits, define representatives for them and
take advantage of this fact to compute them in an easier way than the usual purely exhaustive way.





\end{abstract}
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%Hadamard matrix, cocyclic matrix.\end{IEEEkeywords}






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%\section{Introduction}
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\section{Introduction}

%\IEEEPARstart{H}{
Hadamard matrices are $n\times n$ square matrices $H$ with entries in $\{1, -1\}$ such that every pair of rows (respectively, columns)
are orthogonal, that is, $H  H^T=nI_n$.

It is easy to check that if $H$ is a Hadamard matrix, then $n$ is 1, 2 or a multiple of 4. The Hadamard Conjecture asserts that there exist Hadamard matrices for every size $n$ multiple of 4.

A cocyclic matrix $M_f$ over a group $G$ is a matrix $M_f=(f(g,h))$, for $f$ being a $2$-cocycle over $G$, that is, a function $f:G \times G \rightarrow \{1,-1\}$ such that for every $a,b,c \in G$, $f(a,b)\cdot f(ab,c)\cdot f(a,bc) \cdot f(b,c)=1$. They were first noticed in \cite{HdL93,HdL95}, and since then, many Hadamard matrices have been checked to be cocyclic. For more on cocyclic matrices, the interested reader is referred to \cite{Hor07} and the references there cited.

In  \cite{AGG11}, a characterization of $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices was described, depending on the notions of {\em distributions}, {\em ingredients} and {\em recipes}, which we do not reproduce here, since they are far from  the scope of the present paper. In the process, the $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices were expressed as the product of some elementary coboundaries and a matrix $R$ coming from representative cocycles. The use of the complete set of elementary coboundaries instead of just a basis for coboundaries implied that the same $\Z _t \times \Z_2^2$-cocyclic Hadamard matrix could be expressed in 8 different ways (see \cite{AGG11} for details). This way, the full set of $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices was determined, for $3 \leq t \leq 13$.

In this paper we will introduce what we call {\it diagrams}, a visual representation of the coboundaries which define a $\Z _t \times \Z_2^2$-cocyclic matrix.


Then, we will study four different operations  on the set of coboundary matrices over $\Z_t \times \Z_2^2$: complements, rotations, swappings and dilatations. In particular, these operations extend to operations over $\Z_t \times \Z_2^2$-cocyclic matrices, which will be proved to preserve the Hadamard character of the matrices. For commodity, these operations will be defined on the full set of elementary
coboundaries over $\Z_t \times \Z_2^2$ (not just a basis), so that after applying some of them, the $\Z_t \times \Z_2^2$-cocyclic matrices obtained so far might be expressed in any of the eight possible ways described in \cite{AGG11}. Anyway,
there is no imprecission, as we will see.  These operations partition the set of $\Z_t \times \Z_2^2$-cocyclic matrices into disjoint orbits, which can be easily computed once one element is known. Among all the elements
of an orbit, a representative can be chosen, in a standard way that will be precised.






Finally, by applying these ideas to the task of searching for cocyclic Hadamard matrices over $\Z _t \times \Z_2^2$,
we have been able to extend the table in p. 132 in \cite{BH95} for values of $t$ in the range $3 \leq t \leq 23$, and to
explain the fact that the set of symmetric Williamson type Hadamard matrices is in proportion $\frac{1}{t}$ with respect to the
full set of  $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices.

A preliminary version of this work can be found in \cite{Gue11}.

We organize the paper as follows. Section II is dedicated to the description of
diagrams, after a discussion about the convenience of using the
whole set of coboundaries instead of the basis, in order to
represent $\Z _t \times \Z_2^2$-cocyclic matrices. Section III
defines the four Hadamard preserving operations, which have a nice
interpretation in terms of diagrams. These operations split the set
of Hadamard matrices into disjoint orbits, which can be generated from any of their elements (for instance by its
representative). Last section includes some final remarks and further work.



\section{On basis, generators and diagrams}

In this section we introduce some elementary notations, and give the notion of {\em diagram}, a very useful presentation of a $\Z _t \times \Z_2^2$-cocyclic matrix. In particular, the notion of symmetry in a diagram (see Definition \ref{defsimetria}) will lead to a fast Hadamard test for $\Z _t \times \Z_2^2$-cocyclic matrices, which we state in Theorem \ref{teorema1}.


In \cite{AAFR09} a basis for 2-cocycles over $\Z_t \times \Z_2^2$ is described.
This  basis ${\cal B}=\{\partial _2,\ldots, \partial _{4t-2},\beta_1,\beta_2,\gamma\}$ for 2-cocycles consists
of $4t-3$ coboundaries $\partial _k$, two cocycles $\beta_i$ coming from inflation and one cocycle $\gamma$ coming from
transgression.

In these circumstances, every 2-cocycle over $G$ admits an unique representation as a product of the generators in ${\cal B}$,
$f=f_1^{\alpha_1}\cdots f_k^{\alpha_k}$, $\alpha _i \in \{0,1\}$. The tuple $(\alpha_1,\ldots,\alpha_k)_{\cal B}$ defines the coordinates
of $f$ with regards to ${\cal B}$. Accordingly, every cocyclic matrix $M_f=(f(g_i,g_j))$ for $f=(\alpha_1,\ldots,\alpha_k)_{\cal B}$
admits an unique decomposition as the Hadamard (pointwise) product $M_f=M_{f_1}^{\alpha_1} \cdots  M_{f_k}^{\alpha_k}$.

As usual, $\partial _i$ refers to the coboundary associated to the $i^{th}$-element in $G$.
Let us recall that, once we fixed the basis ${\cal B}$ of coboundaries,  $\{
M_{\partial_{4t-2}},M_{\partial_{4t-3}}, \ldots M_{\partial_3},
M_{\partial_2} \}$, each of the three dropped coboundaries  $ \{ M_{\partial_1},
M_{\partial_{4t-1}}, M_{\partial_{4t}}\}$ can be expressed in terms of the basis, namely :

\begin{itemize}
\item $ \displaystyle M_{\partial_{1}} =  - \prod_{i=1}^{t-1} M_{\partial_{4i+1}} \prod_{i=0}^{t-1}
M_{\partial_{4i+2}}.$

\item  $ \displaystyle M_{\partial_{4t-1}} = \prod_{i=0}^{t-2} M_{\partial_{4i+3}} \prod_{i=0}^{t-1}
M_{\partial_{4i+2}}.$

\item $ \displaystyle M_{\partial_{4t}} = \prod_{i=1}^{t-1} M_{\partial_{4i}} \prod_{i=0}^{t-1}
M_{\partial_{4i+2}}.$


\end{itemize}

From now on, we will denote the coboundary matrix corresponding to the element $\partial_i$ by its number $i$,
and assume that every cocyclic matrix $M$ is obtained as a pointwise product of some coboundaries and the representative
matrix $R= M_{\beta_1} M_{\beta_2} M_\gamma$ (see \cite{BH95} for details).



Depending on whether we use (or not) each of these three coboundaries, we will have
eight different expressions for every cocyclic matrix (these eight expressions are related by means of
substitutions).

For  $t=7$,  Table \ref{reparto5} shows the eight
different expressions of the same Hadamard matrix. First of them is
the expression with respect to the basis, not using any of the three dropped coboundaries.

\begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{$t=7$. Eight expressions for the same Hadamard matrix}\label{reparto5}\centering
$\begin{array}{|c|} \hline
  \{    \{ 14,10,6 \}, \{ 11 \}, \{20,12,4  \}, \{ 25,21,9  \} \}  \\
\hline
 \{    \{ 26,22,18,2 \}, \{ 27, 23,19,15,7,3 \},  \{20,12,4  \}, \{ 25,21,9   \} \}   \\
\hline
 \{    \{ 26,22,18,2 \}, \{ 11 \},  \{28, 24,16,8  \}, \{ 25,21,9   \} \}   \\
\hline
 \{    \{ 26,22,18,2 \}, \{ 11 \},  \{20,12,4  \}, \{ 17, 13, 5,1   \} \}   \\
\hline
   \{    \{ 14,10,6 \}, \{ 27, 23,19,15,7,3 \}, \{28, 24,16,8  \}, \{ 25,21,9  \} \}  \\
\hline
\{  \{ 14,10, 6 \}, \{ 27, 23, 19,15,7,3  \}, \{20,12,4 \}, \{ 17, 13,5,1    \}   \}   \\
\hline
  \{  \{ 14,10, 6 \}, \{ 11 \}, \{  28,24,16,8    \}, \{  17,13,5,1  \}   \}    \\
\hline
 \{  \{ 26,22,18,2 \}, \{ 27, 23, 19,15,7,3 \}, \{28,24,16,8  \}, \{ 17,13,5,1 \}  \} \\
\hline
\end{array}$
\end{table}


%\section{Diagrams}

Cocyclic matrices over  $\Z_t \times \Z_2^2$ can be visualized by
diagrams which represent the coboundaries which appear in the
expression of the matrix.


\begin{definition}
Given a pointwise product of coboundaries  $ M_{\partial_{d_1}}
\ldots M_{\partial_{d_k}} R$ defining a cocyclic matrix over $\Z_t
\times \Z_2^2$,  its  {\it{diagram}} is a  $4 \times t$ matrix, such
that  $a_{ij}, \ 1 \leq i \leq 4, \ 1\leq j \leq t$ is
$\times$, ($ a_{ij} = \times$)  if  $ 4t - 4(j-1)-3 + i \mod   4t
\in \{ d_1, \ldots , d_k  \}$
 and empty elsewhere.
\end{definition}

\begin{remark}
The definition of diagram has to do with the expression of the
matrix in terms of the coboundaries, so every cocyclic matrix over $\Z_t \times \Z_2^2$ has eight
different diagrams.
\end{remark}

For instance, one Hadamard matrix for $t=7$ is given by the following coboundaries:
$$\{ \{14, 10, 6\}, \{11\}, \{20, 12, 4
\}, \{25, 21, 9 \} \}$$

A presentation of this subset of coboundaries as a $4 \times t$ matrix, such that coboundaries $\mod i$ are placed at row $i-1$ is:
$$\left|
    \begin{array}{ccccccc}
      26 & 22 & 18 & {\bf {14}} & {\bf {10}} & {\bf {6}} & 2 \\
      27 & 23 & 19 & 15 & {\bf {11}} & 7 & 3 \\
      28 & 24 & {\bf {20}} & 16 & {\bf {12}} & 8 & {\bf {4}} \\
      {\bf {25}} & {\bf {21}} & 17 & 13 & {\bf {9}} & 5 & 1 \\
    \end{array}
  \right|$$

In short, using $\times$ instead of the subindexes of the coboundaries, we get the diagram
$$\left|
    \begin{array}{ccccccc}
        - & - & - & \times & \times & \times & - \\
       - & - & - & -  & \times & - & - \\
       - & - & \times & - & \times &  - & \times  \\
       \times & \times & - & - & \times & - & -\\
    \end{array}
  \right|$$

Diagrams are a very useful tool. This presentation of the
coboundaries allows us to read easily  the adjacency conditions,
the number of paths, their length and productivity, as introduced in \cite{AGG11}.

A set $\{M_{\partial_{i_j}}: \; 1 \leq j \leq w\}$ of generalized coboundary matrices defines a {\em $n$-walk} if these
matrices may be ordered in a sequence $(M_{l_1},\ldots,M_{l_w})$ so that consecutive matrices $M_{l_i}$ and $M_{l_{i+1}}$ share a negative entry at the
$n^{th}$-row, precisely at the position $(n,l_{i+1})$, $1 \leq i \leq w-1$. Such a walk is called a {\em path} if the initial (equivalently, the final) matrix shares a $-1$ entry with a generalized
coboundary matrix which is not in the walk itself, and a {\em cycle} otherwise.

With this notation at hand, the summation of the $n^{th}$-row of a matrix $M_{\partial_{i_1}}\ldots M_{\partial_{i_w}} \cdot R$ is zero if and only if $2c_n+r_n-2I_n=2t$, where $c_n$  is the number of maximal $n$-paths in $\{M_{\partial_{i_1}},\ldots,M_{\partial_{i_w}}\}$, $r_n$ is the number of
$-1$s in the $n^{th}$-row of $R$ and $I_n$ is the number of positions in which $R$ and $M_{\partial_{i_1}}\ldots M_{\partial_{i_w}}$ share a common $-1$
in their $n^{th}$-row (obviously, $0\leq I_n \leq r_n$).

At this point, it is important to notice that paths and intersections give a Hadamard test for cocyclic matrices, equivalent to the usual cocyclic Hadamard test of \cite{BH95}, which claims that a cocyclic matrix is Hadamard if and only if the summation of each row (but the first) is zero. When this test is applied to $\Z _t \times \Z_2^2$-cocyclic matrices, it suffices to check whether the summation of each of the rows from 5 to $2t+2$ is 0 (see  \cite{AAFR08,AGG11} for details).

\begin{corollary}\cite{AGG11}\label{corolario1} In particular, for  $\Z _t \times \Z_2^2$-cocyclic matrices:

\begin{enumerate}

\item the summation of a row $n \equiv 1 \mod 4$ is zero if and only if \begin{equation}\label{filahadamard1} c_n=t, \end{equation}

\item the summation of a row $n \equiv 0,2,3 \mod 4$ is zero if and only if \begin{equation}\label{filahadamard2} c_n=I_n. \end{equation}

\end{enumerate}
\end{corollary}


Every row of a diagram represents one  $5$-cycle.
 Actually, the diagram is not a matrix, but a cylinder, the
first and last columns being adjacent.
The paths in rows  not congruent to $1$ can be read by alternating
pair of rows in the diagram. The productivity of the even paths comes from the
congruency class module $4$ of its first (or last) coboundary.
This can be easily checked in any diagram.



% Every row of the diagram represents one  $5$-cocycle. The number
% of $5$-paths in each component can be computed by reading each row
% of the diagram. Actually, the diagram is not a matrix, but a cylinder, being the
% first and last columns adjacent.
%
%
% The cocycles in rows  $6$, $7$ and  $8$ can be read by alternating
% pair of rows in the diagram. For instance one of the $6$-cycles is
% obtained by alternating  (zig-zag) first and fourth rows, and the
% other is second/third. $7$-cocycles are first/third and
% second/fourh and  $8$-cocycles are first/second and third/fourth.
%
% The productivity/improductivity of the even paths comes from the
% congruency class module $4$ of its first (or last) coboundary.
% This can be easily checked in the diagram, depending on the row
% where the path starts.
%
% In the previous example, at the row $8$ we have  $3$ paths of length  $1$, $ (10), (12), (9)$,
% one of length $3$, $(14, 11,6)$, and two of
% length $2$, one of them productive $ (4,25) $ (gives two intersections) and the other
% improductive $(28,21)$ (gives no intersection).
%
%
%
% %\begin{remark}
% After the exhaustive searching for $t<15$, all diagrams for
% Hadamard matrices present certain symmetry, for instance, the
% diagram for matrix above is symmetric with respect to the fifth
% column.
% %\end{remark}


\begin{definition}\label{defsimetria}
A diagram associated to a pointwise product of coboundaries  $M =
M_{\partial_1} \ldots M_{\partial_k} R$ is called {\it{symmetric}} if the
$\times$ are symmetric with respect to a column. If one of the
diagrams representing a cocyclic matrix is symmetric, so are all
other diagrams (they are obtained by complementing some of the
rows, and symmetry is preserved). We will say, by extension, that
the cocyclic matrix presents symmetry (the matrix is not
symmetric, the diagram is).
\end{definition}


\begin{theorem} \label{teorema1}
If a cocyclic matrix over   $\Z_t \times \Z_2^2$ is represented by
symmetric diagrams and has exactly $t$ paths in rows congruent to
$1 \mod  4$, then it is Hadamard.
\end{theorem}

\proof

Taking into account Corollary \ref{corolario1}, we have to prove that $c=I$ for rows $6$, $7$, $8$ and congruent.
Odd paths give one intersection, and the only thing to prove is
that even paths are equally distributed between productive and
improductive. Even paths come in pairs (they do not use any of the coboundaries on the
symmetry axis) and the productivity of each path is the opposite
of its symmetric.

\finproof



\section{Operations}

In this section we will study four different operations on the set of coboundary matrices over $\Z_t \times \Z_2^2$: complements, rotations, swappings and dilatations. In particular, these operations extend to operations over $\Z_t \times \Z_2^2$-cocyclic matrices, which will be proved to preserve the Hadamard character of the matrices. These operations partition the set of $\Z_t \times \Z_2^2$-cocyclic matrices into disjoint orbits, which can be easily computed once one element is known. Among all the elements
of an orbit, a representative can be chosen, in a standard way that will be precised.

\subsection{Complements. The group $\Z_2$}



Given a set of coboundaries, $t$, and one of the four congruency
classes  $\mod  4$, for instance $i$-th, $1 \leq i \leq 4$, we
can consider the cocyclic matrix defined by the set of
coboundaries obtained by substituting the subset of coboundaries belonging
to the $i$-th congruency by its complement.




\begin{definition}
Let   $ \{ \{c_{2 j_2} \},\{ c_{3 j_3}\},\{c_{4 j_4}\},
\{c_{1 j_1} \} \}$ with $j_k \subseteq [k]_4$, be a set of coboundaries. The {\it{ complement in the
$i$-th component }}, $1 \leq i \leq 4$, denoted $ {\textsc{C}_i} (\{ \{c_{2 j_2} \},\{ c_{3 j_3}\},\{c_{4 j_4}\},
\{c_{1 j_1} \} \} )$, is the union of the complement of  $\{ c_{i j_i}\}$ in the  $i$-th component and the rest of the
initial coboundaries.
\end{definition}


For instance , if we choose  $i=2$:
$$  {\textsc{C}_2}  ( \{ \{14, 10, 6\}, \{11 \}, \{28, 12, 4\}, \{25, 21, 9\} \}) =  $$
$$ =  \{ \{26, 22, 18, 2 \}, \{11 \}, \{28, 12, 4\}, \{25, 21, 9\} \} $$

In  terms of diagrams:

$$ {\textsc{C}_2} \left|
    \begin{array}{ccccccc}
       -&  - & - &  \times & \times  & \times  & - \\
       -& - & - &  - &  \times & - & - \\
       -& - &  \times & - & \times & - &  \times  \\
      \times & \times &  - &  - & \times  & - & -  \\
    \end{array}
  \right|=  $$

  $$\left|
    \begin{array}{ccccccc}
       \times & \times  & \times &  - & -  & -  & \times \\
       -& - & - &  - &  \times & - & - \\
       -& - &  \times & - & \times & - &  \times  \\
      \times & \times &  - &  - & \times  & - & -  \\
    \end{array}
  \right|$$




\begin{lemma} If we express the matrices with respect to the basis
of coboundaries, then the only complement to compute is the
complement with respect to the component congruent to $2$.
\end{lemma}

\proof

If we consider any other complement, there will appear one of the
coboundaries  $1$, $4t-1$ or $t$. Substitution of the expression
of this coboundary gives us the complement with respect to the
congruency $2$.

\finproof


If two different sets of coboundaries give the same cocyclic matrix
their complements define the same matrix. It suffices to compute the
image of the  different  expressions for a cocyclic matrix and
check that we get the  different expressions for its image. So,
there is no imprecission when we say the {\it complement matrix}
of a cocyclic matrix over $\Z_t \times \Z_2^2$.





\begin{theorem}
 If a set a coboundaries define a Hadamard matrix, so does its complement.
\end{theorem}

  \proof

It suffices to check that the hypotheses of Theorem \ref{teorema1} are satisfied.

The complement of a set of paths in a cycle is another set  with
the same number of paths. So the Hadamard condition in rows
congruent to $1$ is preserved.

For the other rows, just observe that the complement preserves
symmetry in the diagram.

\finproof


\begin{remark}
Note that the operation {\it complement} modifies the number of
coboundaries, substituting  the $k_0$ coboundaries congruent to $2
\mod   4$ by $t-k_0$ coboundaries.
\end{remark}

This operation can be observed as an action of the group $\Z_2$
over the set of $\Z_t \times \Z_2^2$-cocyclic matrices.
For every Hadamard cocyclic matrix over  $\Z_t \times \Z_2^2$, we
can consider its orbit. This orbit has exactly $2$ elements.

\subsection{Rotations. The group $\Z_t$}

Given that the Hadamard condition for a cocyclic matrix over $\Z_t
\times \Z_2^2$ depends on the conditions of position and adjacency
of the coboundaries defining such matrix, one natural idea is
rotating the positions of the coboundaries.


\begin{definition}
Let   $ \{ \{c_{2 j_2} \},\{ c_{3 j_3}\},\{c_{4 j_4}\}, \{c_{1
j_1} \} \}$ with $j_k \subseteq [k]_4$, be a set of coboundaries.
The {\it{$i$-rotated set}},  $0 \leq i \leq t-1$ of this set of
coboundaries, denoted by $\textsc{T}_i ( \{ \{c_{2 j_2} \},\{ c_{3
j_3}\},\{c_{4 j_4}\}, \{c_{1 j_1} \} \})$ is the set:
$$  \{ \{c_{2 j_2 -4i} \},\{c_{3 j_3 -4i }\}, \{c_{4 j_4 -4i} \}, \{ c_{1 j_1-4i}\} \} $$
where additions are $\mod  4t$.
\end{definition}



For instance, for  $i=2$
$$ \textsc{T}_2 ( \{ \{14, 10, 6\}, \{11\}, \{20, 12, 4\}, \{25, 21, 9 \} \} )= $$
$$ =\{ \{ 6, 2, 26 \}, \{3\}, \{12, 4, 24\}, \{ 17, 13, 1  \} \}$$

% Expresado respecto de la base $$\{ (34, 22,6), ( 35,31,27,23,11,7,3), (40, 32, 28,
%20), (41, 33,29,25,21,13) \} $$
In diagrams:

$$\textsc{T}_2 \left|
    \begin{array}{ccccccc}
       -&  - & - &  \times & \times & \times  & - \\
       -& - & - & - & \times  & - & -  \\
       -& - &  \times & - & \times &  - & \times  \\
      \times & \times & - &  - & \times  & - & -  \\
    \end{array}
  \right|=  $$   $$\left|
    \begin{array}{ccccccc}
         \times & - & - & - & - & \times & \times  \\
       -&  - & -  & - & -   & - & \times\\
       - &  \times & - &  - & \times  & - & \times \\
      - &  - & \times  & \times & -  & - & \times  \\
    \end{array}
  \right|$$



The  $i$-rotation operation moves the marked positions  $i$ places
to the right.

As in the complement operation, $i$-rotation over each of the
eight expressions for a Hadamard matrix gives each of the eight
expressions of the same cocyclic Hadamard matrix, so we are
rigourous when speaking of the  $i$-rotated of a matrix.

This operation can be observed as an action of the group $\Z_t$
over the set of $\Z_t \times \Z_2^2$-cocyclic matrices. The element  $i$,
 $0 \leq i \leq t-1$ acts on every cocyclic matrix by substracting  $4i \mod   4t$
 to each of the coboundaries defining the matrix.




\begin{theorem}
The $i$-rotated set of a set defining a cocyclic Hadamard matrix
over $\Z_t \times \Z_2^2$ defines a Hadamard matrix too.
\end{theorem}

\proof

%Once again, it suffices to check whether the hypotheses of Theorem \ref{teorema1} are satisfied.

Rotations preserve the relative positions of the coboundaries at every
row and, thus, the number, length and productivitiy of the paths.

\finproof

\begin{remark}
There is no need to consider symmetry condition for this proof.
\end{remark}


For every Hadamard cocyclic matrix over  $\Z_t \times \Z_2^2$, we
can consider its orbit. This orbit has exactly $t$ elements.

\begin{remark}
Notice that the complement and rotation operations commute.
\end{remark}


\subsection{Swappings. The group $S_4$}

The word  {\it{swapping}} explains clearly the effect of this
operation on a set of coboundaries. Adjacency conditions are
permuted for some of the four components.

We will denote the set of coboundaries:
$$ \{ \{c_{2 j_2}\},\{c_{3 j_3}\},\{c_{4 j_4}\}, \{c_{1 j_1}\} \}$$by $ \{ {\bf {c_2,c_3,c_4,c_1}} \}$,
and the expression $ {\bf {c_j + k }}$ denotes the set of coboundaries obtained by adding $k$ to each of ${\bf {c_j}}$.

\begin{definition}
For the set of coboundaries  $  \{ {\bf {c_2,c_3,c_4,c_1}} \} $ we define the  {\it{swapping operations}}:
\begin{itemize}
\item $s_{23} ( \{ {\bf {c_2,c_3,c_4,c_1}} \} ) = \{ {\bf {c_3 -1,c_2+1,c_4,c_1}} \}. $
\item $s_{24} ( \{ {\bf {c_2,c_3,c_4,c_1}} \} ) = \{ {\bf {c_4-2,c_3,c_2+2,c_1}} \}. $
\item $s_{21} ( \{ {\bf {c_2,c_3,c_4,c_1}} \} ) = \{ {\bf {c_1+1,c_3,c_4,c_2-1}} \}. $
\item $s_{34} ( \{ {\bf {c_2,c_3,c_4,c_1}} \} ) = \{ {\bf {c_2,c_4-1,c_3+1,c_1}} \}. $
\item $s_{31} ( \{ {\bf {c_2,c_3,c_4,c_1}} \} ) = \{ {\bf {c_2,c_1+2,c_4,c_3-2}} \}. $
\item $s_{41} ( \{ {\bf {c_2,c_3,c_4,c_1}} \} ) = \{ {\bf {c_2,c_3,c_1+3,c_4-3}} \}. $


\end{itemize}
\end{definition}



Although we have defined six operations, any composition of them
can be considered, obtaining one of the possible $24$ permutations
on the rows of the diagram.

\begin{theorem}
If $ \{ {\bf {c_2,c_3,c_4,c_1}} \} $ is a set of coboundaries
defining a Hadamard cocyclic matrix over $\Z_t \times \Z_2^2$,
then  $s_{ij} ( \{ {\bf {c_2,c_3,c_4,c_1}} \} )$, $1 \leq i < j
\leq 4$ is also a Hadamard cocyclic matrix over $\Z_t \times
\Z_2^2$.
\end{theorem}

\proof

Once again, it suffices to check whether the hypotheses of Theorem \ref{teorema1} are satisfied.

On one hand, any swapping preserves the number of paths in rows congruent to $1$.

On the other hand, any permutation of the rows in the diagram
preserves symmetry.

\finproof


\begin{remark}
This result can be proved without using the symmetry condition,
one only need to assume that the productive and improductive paths
are equally distributed among the two subsets of components
defined in each row, attending to the congruency
  $\mod   4$ of the coboundaries.
\end{remark}

This gives us an action of $S_4$ on diagrams, which commute with complement and rotations. Depending on
the diagram (i.e. on the size of the permutation group of its rows), the orbit can have less than $24$ elements (actually, 1,4,6,12 or 24).

\subsection{Dilatations. The group $\Z_t^{\star}$}



As we have seen, the  Hadamard character of a cocyclic matrix over
$\Z_t \times \Z_2^2$ is determined by the position and adjacency
of the coboundaries which define it. This is the reason because of rotating the rows of a diagram preserves the Hadamard character of the  corresponding $\Z_t \times \Z_2^2$-cocyclic matrix. Now we are interested in other kind of geometric transformations: homothecies. %and lower or equal to $2t-1$.



\begin{definition}
Fixed  $t$ and given a set of coboundaries $S$ determining a
diagram with columns $ ({\cal{C}}_{t},\cdots ,{\cal{C}}_1)$, the {\it{$r$-th
dilatation}} applied to $S$,
with $r \in \Z / \Z_t^\star$, denoted by $V_r (S)$, is the set corresponding to the image of the
diagram under the homothecy with ratio $r$ and
center the column placed at the right hand-side of the diagram.
\end{definition}

For instance, for  $i=2$
$$ \textsc{V}_2 ( \{ \{14, 10, 6\}, \{11\}, \{20, 12, 4\}, \{25, 21, 9 \} \} )= $$
$$ =\{ \{ 26, 18, 10 \}, \{19\}, \{20, 8, 4\}, \{ 21, 17, 13\} \}$$

% Expresado respecto de la base $$\{ (34, 22,6), ( 35,31,27,23,11,7,3), (40, 32, 28,
%20), (41, 33,29,25,21,13) \} $$
In diagrams:

$$\textsc{V}_2 \left|
    \begin{array}{ccccccc}
       -&  - & - &  \times & \times & \times  & - \\
       -& - & - & - & \times  & - & -  \\
       -& - &  \times & - & \times &  - & \times  \\
      \times & \times & - &  - & \times  & - & -  \\
    \end{array}
  \right|=  $$   $$\left|
    \begin{array}{ccccccc}
           \times & - & \times  & - & \times  & -   & -\\
       -&  - & \times  & - & -   & -  & - \\
       - & -  & \times &  - & - & \times  &   \times \\
      - &  \times  & \times & \times  & -   & - & -  \\
    \end{array}
  \right|$$

As the column located at the right-hand side corresponds to the coboundaries $2,3,4,1$, a formula for $V_r$ is given by $V_r(\partial _k)=\partial _h$, for \begin{equation} \label{dilatacion} h=4 \left[ \frac{k - [k]_4}{4} \cdot r\right]_t+[k]_4. \end{equation} Here we adopt the notation $[m]_n$ for $m \mod n$ for brevity.

\begin{remark}
Notice that a dilatation $V_r$ defines a bijection over the set of $\Z_t \times \Z_2^2$-coboundaries if and only if $r \perp t$, that is, $r\in \Z_t^*$.
\end{remark}

\begin{theorem}
Dilatations $V_r$ with $r \perp t$ preserve the Hadamard character of a $\Z_t \times \Z_2^2$-cocyclic
matrix.
\end{theorem}

\proof

It suffices to check whether the hypotheses of Theorem \ref{teorema1} are satisfied.

Dilatations define some permutation on the number
of paths defined in each row congruent to $1$, so the total number
of paths remains equal to $t$.

Since the image diagram is symmetric as well, the result holds.

\finproof


\begin{remark} The size of the orbit of a symmetric $\Z_t \times \Z_2^2$-cocyclic  matrix under the action of dilatations is, at most,  ${\phi(t)}$, being $\phi$ the Euler function.
\end{remark}


\begin{remark}
Depending on the diagrams, the image under dilatations can coincide with the image
under some of the previously defined operations; for instance the action on the previous diagram of the dilatation
$V_2$ coincides with the action of the composition $s_{14} s_{24} T_5$.
\end{remark}


\subsection{Orbits and representatives}



\begin{definition}
The {\it{total orbit}} of a cocyclic Hadamard matrix over $\Z_t
\times \Z_2^2$ is the union of all orbits under the action of
complement, rotations, swappings and dilatations.
\end{definition}

\begin{definition}
A {\it representative} of a total orbit is a set of coboundaries associated to
a diagram, with minimal number of coboundaries,  symmetric with respect to
the central column, and with an increasing number of coboundaries on each
row of the diagram.
\end{definition}


Table \ref{tabla3} gives us the description of all Hadamard
matrices obtained in  Table III of \cite{AGG11}, splited in orbits,
which can be generated by its representatives under the action of
the four operations previously defined. The second column shows the size of the orbit
under complement/rotation/swapping/dilatations, when providing new matrices.

% \begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{Orbits of $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices}\label{tabla3}\centering
% $\begin{array}{c|c|c|c} t& distrib.& coboundaries& \# H.\\ \hline
% 3&(1,1,1,0)& \{ \{ \}, \{ 7\}, \{ 8\}, \{ 5\} \}&24\\
%  \hline 5&(3,3,2,2)& \{ \{ 10\}, \{ 11\}, \{ 8, 16\}, \{ 1, 17\} \}&120\\
%  \hline 7&(6,6,6,3)& \{ \{ 14 \}, \{ 11, 15, 19 \} , \\ & & \{ 8, 16, 24 \} , \{ 1, 13, 25 \} \} &336\\
%  &(6,5,5,5)& \{ \{ 10, 18 \}, \{ 11, 19 \} , \\ & & \{ 4,  28 \} , \{ 1, 13, 25 \} \}&504\\
%  \hline 9& (10,10,9,7) &\{ \{ 14, 22 \}, \{ 15, 19, 23 \} , \\ & & \{ 4, 16, 24, 36 \} , \{ 1, 13, 21, 33 \} \}&648\\
% & (10,10,9,7) &\{ \{ 14, 22 \}, \{ 3, 19, 35 \} , \\ & & \{ 12, 16, 24, 28 \} , \{ 1, 9, 25, 33 \} \}&1296\\
% &(9,9,9,9)& \{ \{ 14, 18,  22 \}, \{ 11, 19, 27 \} , \\ & & \{ 8, 20, 32 \} , \{ 1, 17, 33 \} \}&432\\
%  \hline 11&(15,14,14,12)& \{ \{ 18, 22, 26 \}, \{ 7, 15, 31, 39 \} , \\ & & \{ 4, 16, 32, 44 \} , \{  9, 13, 21, 29, 33 \} \}&2640\\
%  \hline 13& (21,21,21,15)& \{ \{  22, 26, 30 \}, \{ 3, 15, 23, 31, 39, 51\} , \\ & & \{ 8, 16, 20, 36, 40, 48 \} ,
% \{ 1, 5, 17, 33, 45, 49 \} \}&3744\\
% &(21,21,18,18)& \{ \{ 18, 22, 30, 34 \}, \{ 7, 15, 39, 47 \} , \\ & & \{ 8, 12,  24, 32, 44, 48\} ,
% \{ 1, 5, 21, 29, 45, 49 \} \}&1872\\
% &(20,20,20,18)& \{ \{ 14, 18, 34, 38 \}, \{ 3, 19, 23, 27, 31 39 \} , \\ & & \{ 8, 24, 28, 32, 44 \} ,
% \{ 5, 17, 25, 33, 45 \} \}&1872\\
% &(20,20,20,18)& \{ \{ 14, 18, 34, 38 \}, \{ 15, 19,  27, 35, 39 \} , \\ & & \{ 4, 12, 28,  44, 52 \} ,
% \{ 1, 9, 25, 41, 49 \} \}&936\\
%  \hline
% \end{array}$
% \end{table}

% \begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{Orbits of $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices}\label{tabla3}\centering
% $\begin{array}{c|c|c|c} t& distrib.& coboundaries& \# H.\\ \hline
% 3&(1,1,1,0)& \{ \{ \}, \{ 7\}, \{ 8\}, \{ 5\} \}&24\\
%  \hline 5&(3,3,2,2)& \{ \{ 10\}, \{ 11\}, \{ 8, 16\}, \{ 1, 17\} \}&120\\
%  \hline 7&(6,6,& \{ \{ 14 \}, \{ 11, 15, 19 \} , \\ &6,3) & \{ 8, 16, 24 \} , \{ 1, 13, 25 \} \} &336\\
% \cline{2-3}   &(6,5,& \{ \{ 10, 18 \}, \{ 11, 19 \} , \\ & 5,5)& \{ 4,  28 \} , \{ 1, 13, 25 \} \}&504\\
%  \hline 9& (10,10, &\{ \{ 14, 22 \}, \{ 15, 19, 23 \} , \\ & 9,7)& \{ 4, 16, 24, 36 \} , \{ 1, 13, 21, 33 \} \}&648\\
% \cline{2-3}  & (10,10, &\{ \{ 14, 22 \}, \{ 3, 19, 35 \} , \\ & 9,7)& \{ 12, 16, 24, 28 \} , \{ 1, 9, 25, 33 \} \}&1296\\
% \cline{2-3}  &(9,9,& \{ \{ 14, 18,  22 \}, \{ 11, 19, 27 \} , \\ &9,9) & \{ 8, 20, 32 \} , \{ 1, 17, 33 \} \}&432\\
%  \hline 11&(15,14,& \{ \{ 18, 22, 26 \}, \{ 7, 15, 31, 39 \} , \\ & 14,12)& \{ 4, 16, 32, 44 \} , \{  9, 13, 21, 29, 33 \} \}&2640\\
%  \hline 13& (21,21,& \{ \{  22, 26, 30 \}, \{ 3, 15, 23, 31, 39, 51\} , \\ & 21,15)& \{ 8, 16, 20, 36, 40, 48 \} ,
% \{ 1, 5, 17, 33, 45, 49 \} \}&3744\\
% \cline{2-3}  &(21,21,& \{ \{ 18, 22, 30, 34 \}, \{ 7, 15, 39, 47 \} , \\ & 18,18)& \{ 8, 12,  24, 32, 44, 48\} ,
% \{ 1, 5, 21, 29, 45, 49 \} \}&1872\\
% \cline{2-3}  & (20,20,& \{ \{ 14, 18, 34, 38 \}, \{ 15, 23, 27, 31, 39 \} , \\ & 20,18)& \{ 8, 24, 28, 32, 48 \} ,
% \{ 5, 17, 25, 33, 45 \} \}&1872\\
% \cline{2-3}  &(20,20,& \{ \{ 14, 18, 34, 38 \}, \{ 15, 19,  27, 35, 39 \} , \\ & 20,18)& \{ 4, 12, 28,  44, 52 \} ,
% \{ 1, 9, 25, 41, 49 \} \}&936\\
%  \hline
% \end{array}$
% \end{table}

\begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{Orbits of $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices}\label{tabla3}\centering
$\begin{array}{c|c|c|} t& orbit & coboundaries\\ \hline
3& 2 \times 3 \times 4 \times 1 & \{ \{ \}, \{ 7\}, \{ 8\}, \{ 5\} \}\\
 \hline 5& 2 \times 5 \times 12 \times 1  & \{ \{ 10\}, \{ 11\}, \{ 8, 16\}, \{ 1, 17\} \}\\
 \hline 7& 2 \times 7 \times & \{ \{ 14 \}, \{ 11, 15, 19 \} , \\ & 24 \times 1 = 336 & \{ 8, 16, 24 \} , \{ 1, 13, 25 \} \} \\
\cline{2-3}   & 2 \times 7 \times & \{ \{ 10, 18 \}, \{ 11, 19 \} , \\ & 12 \times 3 = 504& \{ 4,  28 \} , \{ 1, 13, 25 \} \}\\
 \hline 9& 2 \times 9  \times &\{ \{ 14, 22 \}, \{ 15, 19, 23 \} , \\ & 12 \times 3 = 648 & \{ 4, 16, 24, 36 \} , \{ 1, 13, 21, 33 \} \}\\
\cline{2-3}  & 2 \times 9 \times &\{ \{ 14, 22 \}, \{ 3, 19, 35 \} , \\ & 24 \times 3 = 1296 & \{ 12, 16, 24, 28 \} , \{ 1, 9, 25, 33 \} \}\\
\cline{2-3}  & 2 \times 9 \times & \{ \{ 14, 18,  22 \}, \{ 11, 19, 27 \} , \\ &24 \times 1 = 432 & \{ 8, 20, 32 \} , \{ 1, 17, 33 \} \}\\
 \hline 11& 2 \times 11 \times & \{ \{ 18, 22, 26 \}, \{ 7, 15, 31, 39 \} , \\ & 24 \times 5 = 2640& \{ 4, 16, 32, 44 \} , \{  9, 13, 21, 29, 33 \} \}\\
 \hline 13& 2 \times 13 \times & \{ \{  22, 26, 30 \}, \{ 3, 15, 23, 31, 39, 51\} , \\ & 24 \times 3 = 3744& \{ 8, 16, 20, 36, 40, 48 \} ,
\{ 1, 5, 17, 33, 45, 49 \} \}\\
\cline{2-3}  & 2 \times 13 \times & \{ \{ 18, 22, 30, 34 \}, \{ 7, 15, 39, 47 \} , \\ & 24 \times 3 = 1872 & \{ 8, 12,  24, 32, 44, 48\} ,
\{ 1, 5, 21, 29, 45, 49 \} \} \\
\cline{2-3}  & 2 \times 13 \times & \{ \{ 14, 18, 34, 38 \}, \{ 15, 23, 27, 31, 39 \} , \\ & 24 \times 3 = 1872& \{ 8, 24, 28, 32, 48 \} ,
\{ 5, 17, 25, 33, 45 \} \} \\
\cline{2-3}  & 2 \times 13 \times & \{ \{ 14, 18, 34, 38 \}, \{ 15, 19,  27, 35, 39 \} , \\ & 12 \times 3 = 936 & \{ 4, 12, 28,  44, 52 \} ,
\{ 1, 9, 25, 41, 49 \} \}\\
 \hline
\end{array}$
\end{table}


\section{Application to computations}

There are two special repartitions of coboundaries attending to their congruency,  for
every Hadamard matrix. One is the repartition given by the
expression of the matrix with respect to the basis. And the other
is the minimal repartition (this last one can correspond to the
matrix, or to its complement).

When performing calculations we can restrict to the minimal
repartition, in order to simplify the search, avoiding 15 of the
16 possible repartitions for each distribution (see \cite{AGG11} for details
on repartitions and  distributions, existence and computation).


The  rotation, swapping and dilatations preserve the number
of coboundaries involved. On the other hand, the image of the
Hadamard matrix does not depend on the chosen representation (from
the possible eight). Thus we can obtain the whole orbit of the
matrix under all these operations by only knowing the expression
of the matrix with the minimal repartition, so we can restrict
the searching to this easier case.

Once this is done, we can compute the complement of each of the
obtained matrices, to complete the orbit under all operations.



By following this idea, we have computed the set of cocyclic Hadamard matrices
over $\Z_t \times \Z_2^2$, having a symmetric diagram for $15 \leq t \leq 23$, which
are listed in Tables \ref{sime} and \ref{sime2} in terms of their representatives
and the size of their total orbit under complement/rotation/swapping/dilatations, when providing new matrices.


% \begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{$\Z _t \times \Z_2^2$-cocyclic Hadamard matrices [15,23] }\label{sime}\centering
% $\begin{array}{c|c|c|c} t& distrib.& coboundaries& \# H.\\ \hline
%  \hline 15&(28,28,& \{ \{ 18, 22, 38, 42 \}, \{ 7, 19, 27, 35, 43, 55 \} , \\ &27, 22) & \{  16, 20, 28, 32, 36, 44, 48 \} , \{ 1, 9, 25, 29, 33, 49, 57 \} \} &2880\\
% \cline{2-3}   &(28, 28,& \{ \{ 10, 14, 46, 50 \}, \{ 7, 19, 27, 35, 43, 55 \} , \\ & 27, 22)& \{ 8, 12, 24, 32,40, 52, 56 \} , \{ 1, 5, 9, 29, 49, 53, 57 \} \}&2880\\
% \cline{2-3}   &(28, 28,& \{ \{ 14, 26, 34, 46 \}, \{ 11, 15, 27, 35, 47, 51 \} , \\ & 27, 22)& \{ 16,20, 28, 32, 36, 44, 48 \} , \{ 1, 5, 21, 29, 37,53, 57 \} \}&1440\\
% \cline{2-3}   &(28, 27,& \{ \{ 14, 26, 30, 34, 46 \}, \{ 15, 27, 31, 35, 47 \} , \\ & 25, 25)& \{ 12, 16, 24, 40, 48, 52 \} , \{ 9, 13, 21, 29, 37, 45, 49 \} \}&1440\\
%  \hline 17& (36,35 &\{ \{ 18, 30, 34, 38, 50 \}, \{ 3, 23, 31, 35, 39, 47, 67 \} , \\ & 35, 30)& \{ 16, 20, 28, 36, 44, 52, 56 \} , \{ 1, 13, 21, 25, 41, 45, 53, 65 \} \}&3264\\
% \cline{2-3}  & (36, 35, &\{ \{ 18, 30, 34, 38, 50 \}, \{ 3, 23, 27, 35, 43, 47, 67 \} , \\ & 35, 30)& \{ 4, 16, 28, 36, 44, 56, 68 \} , \{ 13, 17, 21, 29, 37, 45, 49, 53 \} \}&3264\\
% \cline{2-3}  & (36, 35, &\{ \{ 18, 30, 34, 38, 50 \}, \{ 7, 15, 31, 35, 39, 55, 63 \} , \\ & 35, 30)& \{ 12, 20, 24, 36, 48, 52, 60 \} , \{ 5, 9, 13, 21, 45, 53, 57,61 \} \}&3264\\
% \cline{2-3}  & (35, 35, &\{ \{ 18, 26, 30, 38, 42, 50 \}, \{ 3, 19, 31, 39, 51, 67 \} , \\ & 33, 33)& \{ 8, 24, 32, 36, 40, 48, 64 \} , \{ 9, 13, 17, 33, 49, 53, 57 \} \}&3264\\
% \hline 19&(45, 45,& \{ \{ 18, 30, 34, 42, 46, 58 \}, \{ 15, 19, 23, 39, 55, 59, 63 \} , \\ & 42, & \{ 8, 16, 28, 36, 40, 44, 52, 64, 72 \} , \\ & 39)& \{  5, 21, 25, 29, 37, 45, 49, 53, 69 \} \}&8208\\
% \cline{2-3} &(45, 45,& \{ \{ 10, 30, 34, 42, 46, 66 \}, \{ 11, 31, 35, 39, 43, 47, 67 \} , \\ & 42, & \{ 4, 12, 16, 32, 40, 48, 64, 68, 76 \} , \\ & 39)& \{  1, 9, 13, 29, 37, 45, 61, 65, 73 \} \}&4104\\
% \cline{2-3}&(45, 45,& \{ \{ 10, 30, 34, 42, 46, 66 \}, \{ 15, 27, 35, 39, 43, 51, 63 \} , \\ & 42, & \{ 8, 16, 32, 36, 40, 44, 48, 64, 72 \} , \\ & 39)& \{  5, 9, 25, 33, 37, 41, 49, 65, 69 \} \}&8208\\
% \cline{2-3}&(, ,& \{ \{ 6, 10, 34, 42, 66, 70 \}, \{ 11, 23, 27, 35, 43, 51, 55, 67 \} , \\ & , & \{ 8, 16, 20, 36, 44, 60, 64, 72 \} , \\ & )& \{  1, 5, 9, 29, 45, 65, 69, 73 \} \}& --\\
% \cline{2-3}&(, ,& \{ \{ 6, 10, 34, 42, 66, 70 \}, \{ 7, 15, 31, 35, 43, 47, 63, 71 \} , \\ & , & \{ 12, 20, 24, 36, 44, 56, 60, 68 \} , \\ & )& \{  1, 5, 9, 25, 49, 65, 69, 73 \} \}&--\\
% \cline{2-3}&(, ,& \{ \{ 22, 30, 34, 42, 46, 54 \}, \{ 7, 15, 19, 35, 43, 59, 63, 71 \} , \\ & , & \{ 4, 16, 20, 36, 44, 60, 64, 76 \} , \\ & )& \{  9, 13, 17, 25, 49, 57, 61, 65 \} \}&--\\
% \hline 21& (55, 55,& \{ \{  22, 26, 38, 46, 58, 62 \}, \{ 3, 15, 19, 27, 39, 47, 59, 67, 71, 83\} , \\ & 55,45)& \{ 8, 12, 16, 24,  40, 48, 64, 72, 76, 80 \} ,
% \{ 13, 21, 25, 29, 33, 49, 53, 27, 61, 69 \} \}&2016\\
% \cline{2-3}  &(55,54,& \{ \{ 10,  34, 38, 42, 46, 50, 74 \}, \{ 3, 19, 23, 31, 55, 63, 67, 83 \} , \\ & 52, 49)& \{ 12,  24, 28, 36, 44, 52, 60, 64, 76\} ,
% \{ 5, 9, 13, 21, 33, 49, 61, 69, 73, 77 \} \}& 6048\\
% \cline{2-3}  &(55,54,& \{ \{ 18, 26, 38, 42, 46, 58, 66 \}, \{ 19, 23, 31, 35, 51, 55, 63, 67 \} , \\ & 52, 49)& \{ 12,  16, 32, 40, 44, 48, 56, 72, 76\} ,
% \{ 1, 5,  17, 29, 37, 45, 53, 65, 77, 81 \} \}& 6048\\
% \cline{2-3}  &(55,54,& \{ \{ 18, 26, 38, 42, 46, 58, 66 \}, \{ 11, 23, 27, 35, 51, 59, 63, 75 \} , \\ & 52, 49)& \{ 12,  16, 32, 40, 44, 48, 56, 72, 76\} ,
% \{ 1, 5, 9, 25, 37, 45, 57, 73, 77, 81 \} \}& 6048\\
% \cline{2-3}  & (54, 52,& \{ \{ 6, 30, 34, 38, 46, 50, 54, 78 \}, \{ 7, 31, 35, 39, 47, 51, 55, 79 \} , \\ & 52, 52)& \{ 16, 20, 36, 52, 68, 72, 80 \} ,
% \{ 13, 17, 33, 41, 49, 65, 69, 77 \} \}&3024\\
% \cline{2-3}  &(54, 52,& \{ \{ 18, 30, 34, 38, 46, 50, 54, 66 \}, \{ 3, 7, 31, 39, 47, 55, 79, 83 \} , \\ & 52, 52)& \{ 8, 24, 32, 36, 52, 56, 64, 80\} ,
% \{ 1, 17, 21, 29, 41, 53, 61, 65, 81 \} \}& 6048\\
% \cline{2-3}  &(54,52,& \{ \{ 14, 18, 30, 38, 46, 54, 66, 70 \}, \{ 7, 15, 27, 31, 55, 59, 71, 79 \} , \\ & 52, 52)& \{ 8, 12, 20, 24, 64, 68, 72, 80 \} ,
% \{ 13, 21, 25, 37, 41, 45, 57, 61, 69 \} \}&2016\\
% \hline 23& (65, 65,& \{ \{  22, 30, 38, 42, 50, 54, 62, 70 \}, \{ 15, 19, 23, 35, 47, 59, 71, 75, 79\} , \\ & 63, 60)& \{ 12, 20, 28, 40, 44, 52, 56, 68, 76, 84 \} ,
% \{ 9, 13, 25, 29, 33, 57, 61, 65, 77, 81 \} \}&12144\\
%  \hline
% \end{array}$
% \end{table}



% \begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{$\Z _t \times \Z_2^2$-cocyclic Hadamard matrices [15-19] (option) }\label{sime}\centering
% $\begin{array}{c} t/ distrib./ \# H. / coboundaries\\
%  \hline 15 \qquad (28,28,27, 22)  \qquad 2880 \\ \{ \{ 18, 22, 38, 42 \}, \{ 7, 19, 27, 35, 43, 55 \} , \\   \{  16, 20, 28, 32, 36, 44, 48 \} , \{ 1, 9, 25, 29, 33, 49, 57 \} \} \\
% \hline  15 \qquad (28, 28, 27, 22)  \qquad 2880 \\  \{ \{ 10, 14, 46, 50 \}, \{ 7, 19, 27, 35, 43, 55 \} , \\  \{ 8, 12, 24, 32,40, 52, 56 \} , \{ 1, 5, 9, 29, 49, 53, 57 \} \} \\
% \hline
%  15 \qquad (28, 28, 27, 22) \qquad 1440 \\ \{ \{ 14, 26, 34, 46 \}, \{ 11, 15, 27, 35, 47, 51 \} , \\  \{ 16,20, 28, 32, 36, 44, 48 \} , \{ 1, 5, 21, 29, 37,53, 57 \} \}\\
% \hline 15 \qquad (28, 27, 25, 25) \qquad 1440 \\ \{ \{ 14, 26, 30, 34, 46 \}, \{ 15, 27, 31, 35, 47 \} , \\  \{ 12, 16, 24, 40, 48, 52 \} , \{ 9, 13, 21, 29, 37, 45, 49 \} \}\\
% \hline
% \hline 17 \qquad  (36,35, 35, 30) \qquad 3264 \\ \{ \{ 18, 30, 34, 38, 50 \}, \{ 3, 23, 31, 35, 39, 47, 67 \} , \\ \{ 16, 20, 28, 36, 44, 52, 56 \} , \{ 1, 13, 21, 25, 41, 45, 53, 65 \} \} \\
% \hline 17 \qquad (36, 35, 35, 30) \qquad 3264 \\ \{ \{ 18, 30, 34, 38, 50 \}, \{ 3, 23, 27, 35, 43, 47, 67 \} , \\  \{ 4, 16, 28, 36, 44, 56, 68 \} , \{ 13, 17, 21, 29, 37, 45, 49, 53 \} \}\\
% \hline  \qquad (36, 35, 35, 30) \qquad 3264 \\ \{ \{ 18, 30, 34, 38, 50 \}, \{ 7, 15, 31, 35, 39, 55, 63 \} , \\  \{ 12, 20, 24, 36, 48, 52, 60 \} , \{ 5, 9, 13, 21, 45, 53, 57,61 \} \}\\
% \hline 17 \qquad  (35, 35, 33, 33) \qquad 3264 \\ \{ \{ 18, 26, 30, 38, 42, 50 \}, \{ 3, 19, 31, 39, 51, 67 \} , \\  \{ 8, 24, 32, 36, 40, 48, 64 \} , \{ 9, 13, 17, 33, 49, 53, 57 \} \}\\
% \hline  \hline 19 \qquad (45, 45,42, 39) \qquad 8208 \\ \{ \{ 18, 30, 34, 42, 46, 58 \}, \{ 15, 19, 23, 39, 55, 59, 63 \} , \\ \{ 8, 16, 28, 36, 40, 44, 52, 64, 72 \} , \{  5, 21, 25, 29, 37, 45, 49, 53, 69 \} \} \\
% \hline 19 \qquad (45, 45, 42, 39) \qquad 4104 \\  \{ \{ 10, 30, 34, 42, 46, 66 \}, \{ 11, 31, 35, 39, 43, 47, 67 \} , \\ \{ 4, 12, 16, 32, 40, 48, 64, 68, 76 \} , \{  1, 9, 13, 29, 37, 45, 61, 65, 73 \} \}\\
% \hline 19 \qquad (45, 45, 42, 39) \qquad 8208 \\ \{ \{ 10, 30, 34, 42, 46, 66 \}, \{ 15, 27, 35, 39, 43, 51, 63 \} , \\ \{ 8, 16, 32, 36, 40, 44, 48, 64, 72 \} ,
% \{  5, 9, 25, 33, 37, 41, 49, 65, 69 \} \}\\
% \hline  19 \qquad (44,44,44,39 ) \qquad 2376\\ \{ \{ 6, 10, 34, 42, 66, 70 \}, \{ 11, 23, 27, 35, 43, 51, 55, 67 \} , \\ \{ 8, 16, 20, 36, 44, 60, 64, 72 \} ,  \{  1, 5, 9, 29, 45, 65, 69, 73 \} \} \\
% \hline 19 \qquad (44,44,44,39 ) \qquad 2736\\ \{ \{ 6, 10, 34, 42, 66, 70 \}, \{ 7, 15, 31, 35, 43, 47, 63, 71 \} , \\ \{ 12, 20, 24, 36, 44, 56, 60, 68 \} ,  \{  1, 5, 9, 25, 49, 65, 69, 73 \} \}\\
% \hline 19 \qquad (44,44,44,39 ) \qquad 8208\\ \{ \{ 22, 30, 34, 42, 46, 54 \}, \{ 7, 15, 19, 35, 43, 59, 63, 71 \} , \\  \{ 4, 16, 20, 36, 44, 60, 64, 76 \} , \{  9, 13, 17, 25, 49, 57, 61, 65 \} \}\\
% \hline
% \end{array}$
% \end{table}

\begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{$\Z _t \times \Z_2^2$-cocyclic Hadamard matrices [15-19]}\label{sime}\centering
$\begin{array}{c} t/ orbit / \# H. / coboundaries\\
 \hline 15 \qquad 2 \times 15 \times 24 \times 4  \qquad 2880 \\ \{ \{ 18, 22, 38, 42 \}, \{ 7, 19, 27, 35, 43, 55 \} , \\   \{  16, 20, 28, 32, 36, 44, 48 \} , \{ 1, 9, 25, 29, 33, 49, 57 \} \} \\
\hline  15 \qquad 2 \times 15 \times 24 \times 4   \qquad 2880 \\  \{ \{ 10, 14, 46, 50 \}, \{ 7, 19, 27, 35, 43, 55 \} , \\  \{ 8, 12, 24, 32,40, 52, 56 \} , \{ 1, 5, 9, 29, 49, 53, 57 \} \} \\
\hline
 15 \qquad 2 \times 15 \times 24 \times 2  \qquad 1440 \\ \{ \{ 14, 26, 34, 46 \}, \{ 11, 15, 27, 35, 47, 51 \} , \\  \{ 16,20, 28, 32, 36, 44, 48 \} , \{ 1, 5, 21, 29, 37,53, 57 \} \}\\
\hline 15 \qquad 2 \times 15 \times 12 \times 4  \qquad 1440 \\ \{ \{ 14, 26, 30, 34, 46 \}, \{ 15, 27, 31, 35, 47 \} , \\  \{ 12, 16, 24, 40, 48, 52 \} , \{ 9, 13, 21, 29, 37, 45, 49 \} \}\\
\hline
\hline 17 \qquad 2 \times 17 \times 24 \times 4 \qquad 3264 \\ \{ \{ 18, 30, 34, 38, 50 \}, \{ 3, 23, 31, 35, 39, 47, 67 \} , \\ \{ 16, 20, 28, 36, 44, 52, 56 \} , \{ 1, 13, 21, 25, 41, 45, 53, 65 \} \} \\
\hline 17 \qquad 2 \times 17 \times 24 \times 4 \qquad 3264 \\ \{ \{ 18, 30, 34, 38, 50 \}, \{ 3, 23, 27, 35, 43, 47, 67 \} , \\  \{ 4, 16, 28, 36, 44, 56, 68 \} , \{ 13, 17, 21, 29, 37, 45, 49, 53 \} \}\\
\hline 17 \qquad 2 \times 17 \times 24 \times 4 \qquad 3264 \\ \{ \{ 18, 30, 34, 38, 50 \}, \{ 7, 15, 31, 35, 39, 55, 63 \} , \\  \{ 12, 20, 24, 36, 48, 52, 60 \} , \{ 5, 9, 13, 21, 45, 53, 57,61 \} \}\\
\hline 17 \qquad  2 \times 17 \times 24 \times 4 \qquad 3264 \\ \{ \{ 18, 26, 30, 38, 42, 50 \}, \{ 3, 19, 31, 39, 51, 67 \} , \\  \{ 8, 24, 32, 36, 40, 48, 64 \} , \{ 9, 13, 17, 33, 49, 53, 57 \} \}\\
\hline  \hline 19 \qquad 2 \times 19 \times 24 \times 9 \qquad 8208 \\ \{ \{ 18, 30, 34, 42, 46, 58 \}, \{ 15, 19, 23, 39, 55, 59, 63 \} , \\ \{ 8, 16, 28, 36, 40, 44, 52, 64, 72 \} , \{  5, 21, 25, 29, 37, 45, 49, 53, 69 \} \} \\
\hline 19 \qquad 2 \times 19 \times 12 \times 9 \qquad 4104 \\  \{ \{ 10, 30, 34, 42, 46, 66 \}, \{ 11, 31, 35, 39, 43, 47, 67 \} , \\ \{ 4, 12, 16, 32, 40, 48, 64, 68, 76 \} , \{  1, 9, 13, 29, 37, 45, 61, 65, 73 \} \}\\
\hline 19 \qquad 2 \times 19 \times 24 \times 9 \qquad 8208 \\ \{ \{ 10, 30, 34, 42, 46, 66 \}, \{ 15, 27, 35, 39, 43, 51, 63 \} , \\ \{ 8, 16, 32, 36, 40, 44, 48, 64, 72 \} ,
\{  5, 9, 25, 33, 37, 41, 49, 65, 69 \} \}\\
\hline  19 \qquad 2 \times 19 \times 24 \times 3 \qquad 2376\\ \{ \{ 6, 10, 34, 42, 66, 70 \}, \{ 11, 23, 27, 35, 43, 51, 55, 67 \} , \\ \{ 8, 16, 20, 36, 44, 60, 64, 72 \} ,  \{  1, 5, 9, 29, 45, 65, 69, 73 \} \} \\
\hline 19 \qquad 2 \times 19 \times 24 \times 3 \qquad 2736\\ \{ \{ 6, 10, 34, 42, 66, 70 \}, \{ 7, 15, 31, 35, 43, 47, 63, 71 \} , \\ \{ 12, 20, 24, 36, 44, 56, 60, 68 \} ,  \{  1, 5, 9, 25, 49, 65, 69, 73 \} \}\\
\hline 19 \qquad 2 \times 19 \times 24 \times 9 \qquad 8208\\ \{ \{ 22, 30, 34, 42, 46, 54 \}, \{ 7, 15, 19, 35, 43, 59, 63, 71 \} , \\  \{ 4, 16, 20, 36, 44, 60, 64, 76 \} , \{  9, 13, 17, 25, 49, 57, 61, 65 \} \}\\
\hline
\end{array}$
\end{table}

% \begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{$\Z _t \times \Z_2^2$-cocyclic Hadamard matrices [21-23] (option) }\label{sime2}\centering
% $\begin{array}{c} t/ distrib./ \# H. / coboundaries\\
% \hline 21 \qquad  (55, 55, 55, 45) \qquad 2016 \\ \{ \{  22, 26, 38, 46, 58, 62 \}, \{ 3, 15, 19, 27, 39, 47, 59, 67, 71, 83\} , \\ \{ 8, 12, 16, 24,  40, 48, 64, 72, 76, 80 \} ,
%  \{ 13, 21, 25, 29, 33, 49, 53, 27, 61, 69 \} \}\\
% \hline  21 \qquad (55,54,52, 49) \qquad 6048 \\ \{ \{ 10,  34, 38, 42, 46, 50, 74 \}, \{ 3, 19, 23, 31, 55, 63, 67, 83 \} , \\ \{ 12,  24, 28, 36, 44, 52, 60, 64, 76\} ,
%  \{ 5, 9, 13, 21, 33, 49, 61, 69, 73, 77 \} \}\\
%  \hline 21 \qquad (55,54, 52, 49) \qquad 6048 \\ \{ \{ 18, 26, 38, 42, 46, 58, 66 \}, \{ 19, 23, 31, 35, 51, 55, 63, 67 \} , \\  \{ 12,  16, 32, 40, 44, 48, 56, 72, 76\} ,
%  \{ 1, 5,  17, 29, 37, 45, 53, 65, 77, 81 \} \}\\
% \hline 21 \qquad (55,54, 52, 49) \qquad 6048 \\ \{ \{ 18, 26, 38, 42, 46, 58, 66 \}, \{ 11, 23, 27, 35, 51, 59, 63, 75 \} , \\ \{ 12,  16, 32, 40, 44, 48, 56, 72, 76\} ,
%  \{ 1, 5, 9, 25, 37, 45, 57, 73, 77, 81 \} \} \\
% \hline  21\qquad  (54, 52, 52, 52) \qquad 3024 \\ \{ \{ 6, 30, 34, 38, 46, 50, 54, 78 \}, \{ 7, 31, 35, 39, 47, 51, 55, 79 \} , \\ \{ 16, 20, 36, 52, 68, 72, 80 \} ,
%  \{ 13, 17, 33, 41, 49, 65, 69, 77 \} \}\\
% \hline  21 \qquad (54, 52, 52, 52) \qquad 6048 \\ \{ \{ 18, 30, 34, 38, 46, 50, 54, 66 \}, \{ 3, 7, 31, 39, 47, 55, 79, 83 \} , \\ \{ 8, 24, 32, 36, 52, 56, 64, 80\} ,
%  \{ 1, 17, 21, 29, 41, 53, 61, 65, 81 \} \}\\
%  \hline  21 \qquad (54,52, 52, 52) \qquad 2016 \\ \{ \{ 14, 18, 30, 38, 46, 54, 66, 70 \}, \{ 7, 15, 27, 31, 55, 59, 71, 79 \} , \\  \{ 8, 12, 20, 24, 64, 68, 72, 80 \} ,
%  \{ 13, 21, 25, 37, 41, 45, 57, 61, 69 \} \}\\
% \hline  \hline 23 \qquad (65, 65, 63, 60) \qquad 12144 \\ \{ \{  22, 30, 38, 42, 50, 54, 62, 70 \}, \{ 15, 19, 23, 35, 47, 59, 71, 75, 79\} , \\  \{ 12, 20, 28, 40, 44, 52, 56, 68, 76, 84 \} ,
%  \{ 9, 13, 25, 29, 33, 57, 61, 65, 77, 81 \} \}\\
%   \hline
% \end{array}$
% \end{table}

\begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{$\Z _t \times \Z_2^2$-cocyclic Hadamard matrices [21-23]}\label{sime2}\centering
$\begin{array}{c} t/ orbit/ \# H. / coboundaries\\
\hline 21 \qquad  2 \times 21 \times 24 \times 2 \qquad 2016 \\ \{ \{  22, 26, 38, 46, 58, 62 \}, \{ 3, 15, 19, 27, 39, 47, 59, 67, 71, 83\} , \\ \{ 8, 12, 16, 24,  40, 48, 64, 72, 76, 80 \} ,
 \{ 13, 21, 25, 29, 33, 49, 53, 27, 61, 69 \} \}\\
\hline  21 \qquad 2 \times 21 \times 24 \times 6 \qquad 6048 \\ \{ \{ 10,  34, 38, 42, 46, 50, 74 \}, \{ 3, 19, 23, 31, 55, 63, 67, 83 \} , \\ \{ 12,  24, 28, 36, 44, 52, 60, 64, 76\} ,
 \{ 5, 9, 13, 21, 33, 49, 61, 69, 73, 77 \} \}\\
 \hline 21 \qquad 2 \times 21 \times 24 \times 6 \qquad 6048 \\ \{ \{ 18, 26, 38, 42, 46, 58, 66 \}, \{ 19, 23, 31, 35, 51, 55, 63, 67 \} , \\  \{ 12,  16, 32, 40, 44, 48, 56, 72, 76\} ,
 \{ 1, 5,  17, 29, 37, 45, 53, 65, 77, 81 \} \}\\
\hline 21 \qquad 2 \times 21 \times 24 \times 6 \qquad 6048 \\ \{ \{ 18, 26, 38, 42, 46, 58, 66 \}, \{ 11, 23, 27, 35, 51, 59, 63, 75 \} , \\ \{ 12,  16, 32, 40, 44, 48, 56, 72, 76\} ,
 \{ 1, 5, 9, 25, 37, 45, 57, 73, 77, 81 \} \} \\
\hline  21\qquad  2 \times 21 \times 12 \times 6 \qquad 3024 \\ \{ \{ 6, 30, 34, 38, 46, 50, 54, 78 \}, \{ 7, 31, 35, 39, 47, 51, 55, 79 \} , \\ \{ 16, 20, 36, 52, 68, 72, 80 \} ,
 \{ 13, 17, 33, 41, 49, 65, 69, 77 \} \}\\
\hline  21 \qquad 2 \times 21 \times 24 \times 6 \qquad 6048 \\ \{ \{ 18, 30, 34, 38, 46, 50, 54, 66 \}, \{ 3, 7, 31, 39, 47, 55, 79, 83 \} , \\ \{ 8, 24, 32, 36, 52, 56, 64, 80\} ,
 \{ 1, 17, 21, 29, 41, 53, 61, 65, 81 \} \}\\
 \hline  21 \qquad 2 \times 21 \times 24 \times 2 \qquad 2016 \\ \{ \{ 14, 18, 30, 38, 46, 54, 66, 70 \}, \{ 7, 15, 27, 31, 55, 59, 71, 79 \} , \\  \{ 8, 12, 20, 24, 64, 68, 72, 80 \} ,
 \{ 13, 21, 25, 37, 41, 45, 57, 61, 69 \} \}\\
\hline  \hline 23 \qquad 2 \times 23 \times 24 \times 11 \qquad 12144 \\ \{ \{  22, 30, 38, 42, 50, 54, 62, 70 \}, \{ 15, 19, 23, 35, 47, 59, 71, 75, 79\} , \\  \{ 12, 20, 28, 40, 44, 52, 56, 68, 76, 84 \} ,
 \{ 9, 13, 25, 29, 33, 57, 61, 65, 77, 81 \} \}\\
  \hline
\end{array}$
\end{table}

\subsection{Williamson matrices}


The first row of Table \ref{will} shows the number of Williamson
matrices obtained in \cite{BH95}, together with the total number
of cocyclic Hadamard matrices over $\Z _t \times \Z_2^2$ which we
have computed.

\begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{Williamson / $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices}\label{will}\centering
$\begin{array}{c|c|c|c|c|c|c|c|c|c|}
t & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19  \\
\hline \sharp Will. & 8 & 24 & 120 & 264 & 240 & 648 & 576 & - & -  \\
\sharp H.& 24 & 120 & 840 & 2376 & 2640 & 8424 & 8640 & 13056 & 34200 \\
  \hline
\end{array}$
\end{table}



It can be observed that  the set of symmetric Williamson type
Hadamard matrices is in proportion $\frac{1}{t}$ with respect to
the full set of $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices.



By Lemma 3.4 in \cite{BH95}, any Wiliamson matrix is Hadamard equivalent to a cocyclic matrix over $\Z _t \times \Z_2^2$,
which is $t\times t$ back-circulant by  blocks $4 \times 4$,
\begin{equation}\label{wilcoc} \left( \begin{array}{cccc} W_1 &
\ldots &  \ldots& W_t\\ W_2&_{\cdot}\cdot^{\cdot}&W_t&W_1\\ _{\cdot}\cdot^{\cdot}&
_{\cdot}\cdot^{\cdot}&_{\cdot}\cdot^{\cdot}&_{\cdot}\cdot^{\cdot}\\ W_t&W_1& \ldots&W_{t-1}
\end{array}\right) \end{equation} with  $W_{i+1}=W_{t-i+1}$ for
$1 \leq i \leq t-1$, and  $W_i$ being $$\left(
\begin{array}{rrrr} n_i&x_i&y_i&z_i\\ x_i&-n_i& z_i&-y_i\\
y_i&-z_i&-n_i&x_i\\ z_i&y_i&-x_i&-n_i \end{array}\right).$$

This matrix  (\ref{wilcoc}) is a pointwise product of the
representative matrix $R$ and a certain set of generalized
coboundaries $\{M_{\partial _{d_1}}, \ldots ,
M_{\partial_{d_k}}\}$. The additional condition
$W_{i+1}=W_{t-i+1}$ for $1 \leq i \leq t-1$, leaving  $W_1$ alone
means that the symmetry column in the diagram representing the
matrix (\ref{wilcoc}) is,  precisely, the last one to the right,
which is associated to the coboundaries $\partial _2, \partial _3,
\partial _4, \partial _1$. This fact leads us to conclude that:

\begin{proposition}
Let ${\cal H}$ the set of cocyclic Hadamard matrices over
$\Z_t \times \Z_2^2$, having a symmetric diagram. Then, the set of Williamson matrices of type
(\ref{wilcoc}) is a subset of  ${\cal H}$ of size $\frac{|{\cal H}|}{t}$. Moreover, for each element $H \in
{\cal H}$, one and only one of the rotated matrices $T_iH$, $1 \leq i \leq t$, is a Williamson matrix.
\end{proposition}

\proof

The condition needed tell us that the diagram has to be symmetric
with respect to the last column, which is only possible for one of
the rotated.

\finproof

So, the predicted number of Williamson matrices for $t \in [17,23]$ will be $13056/17=768$, $34200/19=1800$, $31248/21=1488$ and $12144/23=528$,
which is indeed the case (see \cite{Kou}).


\section{Further questions}


After an observation process, some questions come into our minds:


\begin{itemize}

\item All $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices obtained so far have a symmetric diagram up to $t=15$, and we have taken advantage of it
when computing them for $t \in [17,23]$ by restricting ourselves to the symmetric case. Can this symmetry assumption be proved?

\item The result about Williamson matrices allows us to go beyond $t=23$. Actually, taking into account the results in \cite{Kou}, we can express
every Williamson matrix as a pointwise product of coboundaries and
compute the whole orbit of matrices for $t \in [25, 39]$. We show
the results obtained in Table \ref{kou}, after identifying every
matrix obtained in terms of coboundaries, and computing the size
of its orbit under the action of
complement/rotation/swapping/dilatations, when providing
new matrices.

\begin{table}[!t]\renewcommand{\arraystretch}{1.3}\caption{Full orbits from Williamson matrices}\label{kou}\centering
$\begin{array}{c|l|r|}
t & 4t=A^2+B^2+C^2+D^2 & \sharp (comp/rot/swap/dil)   \\
\hline
25 & 100=9^2+3^2+3^2+1^2 & 2*25*24*10=12000  \\
\cline{2-3} & 100=7^2+7^2+ 1^2 + 1^2 &  2*25*12*5=3000  \\
&  &  2*25*24*5=6000  \\
& & 2*25*24*5=6000\\
\cline{2-3} & 100=7^2+5^2+5^2+1^2 & 2*25*24*10=12000 \\
& & 2*25*24*10=12000 \\
& & 2*25*24*5=6000 \\
\cline{2-3} & 100=5^2+5^2+5^2+5^2 & 2*25*24*5=6000 \\
& & 2*25*24*5=6000 \\
& & 2*25*24*5=6000 \\
\hline
27 & 108=9^2+5^2+1^2+1^2 & 2*27*24*9=11664  \\
   & & 2*27*24*9=11664 \\
\cline{2-3} & 108=7^2+7^2+ 3^2 + 1^2 &  2*27*12*9=5832  \\
&  &  2*27*24*9=11664  \\
& & 2*27*24*9=11664 \\
\cline{2-3} & 108=7^2+5^2+5^2+3^2 & 2*27*24*9=11664 \\
\hline
29 & 116=9^2+5^2+3^2+1^2 & 2*29*24*14=19488  \\
\hline
31 & 124=7^2+7^2+ 5^2 + 1^2 &  2*31*24*15=22320  \\
\cline{2-3} & 124=7^2+5^2+5^2+5^2 & 2*31*12*15=11160 \\
\hline
33 & 132=11^2+3^2+1^2+1^2 & 2*33*24*10=15840  \\
\cline{2-3} & 132=9^2+7^2+ 1^2 + 1^2 &  2*33*24*10=15840  \\
\cline{2-3} & 132=9^2+5^2+5^2+1^2 & 2*33*24*10=15840 \\
 & & 2*33*24*10=15840\\
\cline{2-3} & 132= 7^2+7^2+5^2+ 3^2 & 2*33*24*10=15840 \\
\hline
37 & 148=11^2+3^2+3^2+3^2 & 2*37*24*3=5328  \\
\cline{2-3} & 148=9^2+7^2+ 3^2 + 3^2 &  2*37*12*18=15984  \\
\cline{2-3} & 148=7^2+7^2+ 5^2 + 5^2 &  2*37*24*9=15984  \\
&  &  2*37*24*9=15984  \\
\hline 39 & 156=9^2+5^2+5^2+5^2& 2*39*24*4=7488 \\
\hline
41 & 164=9^2+9^2+1^2+1^2 & 2*41*12*5=4920  \\
\hline
43 & 172=7^2+7^2+7^2+5^2 & 2*43*24*7=14448  \\
\hline
45 & 180=9^2+7^2+5^2+5^2 & 2*45*12*12=12960  \\
\hline
49 & 196=9^2+9^2+5^2+3^2 & 2*49*12*21=24696  \\
\hline
51 & 204=11^2+9^2+ 1^2 + 1^2 &  2*51*12*16=19584  \\
\cline{2-3} & 204=11^2+7^2+5^2+3^2 & 2*51*24*16=39168 \\
\hline
55 & 220=11^2+9^2+3^2+3^2 & 2*55*12*20=26400  \\
\hline
57 & 228=9^2+7^2+7^2+7^2 & 2*57*12*18=24624  \\
\hline
61 & 244=11^2+11^2+1^2+1^2 & 2*61*12*15=21960  \\
\hline
63 & 252=11^2+11^2+3^2+1^2 & 2*63*12*6=9072  \\
\hline
\end{array}$
\end{table}


\item Now that we have proved that our operations preserve the Hadamard character, arises the question: Do these operations preserve Hadamard equivalence? Rotations, dilatations and some of the swappings do, because they can be
expressed in terms of the shifting operation defined by Horadam in \cite{Hor07} (this fact will be detailed elsewhere). In addition, the number of orbits coincide with the number of
non equivalent Williamson matrices computed in \cite{Kou}, so  the result is probably true in general.

\end{itemize}

% if have a single appendix:
%\appendix[Proof of the Zonklar Equations]
% or
%\appendix  % for no appendix heading
% do not use \section anymore after \appendix, only \section*
% is possibly needed

% use appendices with more than one appendix
% then use \section to start each appendix
% you must declare a \section before using any
% \subsection or using \label (\appendices by itself
% starts a section numbered zero.)
%




% use section* for acknowledgement
% \section*{Acknowledgment}
%
%
% The authors would like to thank...


%In \cite{BH95}.


%\section*{Acknowledgment}


%The authors would like to thank...


% Can use something like this to put references on a page
% by themselves when using endfloat and the captionsoff option.
%\ifCLASSOPTIONcaptionsoff
%  \newpage
%\fi



% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
\begin{thebibliography}{1}

%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.


\bibitem{AAFR08} V. \'Alvarez, J.A. Armario, M.D. Frau and P. Real, "A system of equations for describing cocyclic Hadamard matrices", {\em J. Comb. Des.}, vol. 16, pp. 276--290, 2008.

\bibitem{AAFR09} V. \'Alvarez, J.A. Armario, M.D. Frau and P. Real, "The homological reduction method for computing cocyclic Hadamard matrices", {\em J. Symb. Comput.}, vol. 44, pp. 558-570, 2009.

\bibitem{AGG11} V. \'{A}lvarez, F. Gudiel and M.B. G\"{u}emes, "On $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices". Preprint. Submitted to journal.

\bibitem{BH95} A. Baliga and K.J. Horadam, "Cocyclic Hadamard matrices over $\Z _t \times \Z _2^2$, {\em Australas. J. Combin.}, vol. 11, pp. 123--134, 1995.

\bibitem{Gue11}
M. B. G\"uemes, ``Cocyclic Hadamard matrices over $\Z _t \times
\Z_2^2$. Description, classification and search. PhD thesis (English translation), University of Seville, 2011.

\bibitem{Hor07} K.J. Horadam, {\em Hadamard matrices and their applications}. Princeton: Princeton University Press,  2007.


\bibitem{HdL93} K.J. Horadam and W. de Launey, "Cocyclic development of designs", {\em J. Algebraic Combin.}, vol. 2, no. 3, pp. 267--290, 1993. Erratum, {\em J. Algebraic Combin.}, vol. 3, pp. 129, 1994.

\bibitem{HdL95} K.J. Horadam and W. de Launey, "Generation of cocyclic Hadamard matrices", in: {\em Proc. Computational Algebra and Number Theory}, Sydney, 1992, in {\em Math. Appl.}, vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 279--290.





\bibitem{Kou} C. Koukouvinos, web page, http://www.math.ntua.gr/~ckoukouv/



%\bibitem{HdL93} K.J. Horadam and W. de Launey, "Cocyclic development of designs", {\em J. Algebraic Combin.}, vol. 2, no. 3, pp. 267--290, 1993. Erratum, {\em J. Algebraic Combin.}, vol. 3, pp. 129, 1994.

%\bibitem{HdL95} K.J. Horadam and W. de Launey, "Generation of cocyclic Hadamard matrices", in: {\em Proc. Computational Algebra and Number Theory}, Sydney, 1992, in {\em Math. Appl.}, vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 279--290.















\end{thebibliography}

% biography section
%
% If you have an EPS/PDF photo (graphicx package needed) extra braces are
% needed around the contents of the optional argument to biography to prevent
% the LaTeX parser from getting confused when it sees the complicated
% \includegraphics command within an optional argument. (You could create
% your own custom macro containing the \includegraphics command to make things
% simpler here.)
%\begin{biography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
% or if you just want to reserve a space for a photo:

%\vfill

%\end{document}


%\vfill{ }

% if you will not have a photo at all:
%\begin{IEEEbiographynophoto}{John Doe}
%Biography text here.
%\end{IEEEbiographynophoto}

% insert where needed to balance the two columns on the last page with
% biographies
%\newpage

%\begin{IEEEbiographynophoto}{Jane Doe}
%Biography text here.
%\end{IEEEbiographynophoto}

% You can push biographies down or up by placing
% a \vfill before or after them. The appropriate
% use of \vfill depends on what kind of text is
% on the last page and whether or not the columns
% are being equalized.

%\vfill

% Can be used to pull up biographies so that the bottom of the last one
% is flush with the other column.
%\enlargethispage{-5in}



% that's all folks
\end{document}

1112.4300.extracted_bib
\begin{thebibliography}{1}

%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.


\bibitem{AAFR08} V. \'Alvarez, J.A. Armario, M.D. Frau and P. Real, "A system of equations for describing cocyclic Hadamard matrices", {\em J. Comb. Des.}, vol. 16, pp. 276--290, 2008.

\bibitem{AAFR09} V. \'Alvarez, J.A. Armario, M.D. Frau and P. Real, "The homological reduction method for computing cocyclic Hadamard matrices", {\em J. Symb. Comput.}, vol. 44, pp. 558-570, 2009.

\bibitem{AGG11} V. \'{A}lvarez, F. Gudiel and M.B. G\"{u}emes, "On $\Z _t \times \Z_2^2$-cocyclic Hadamard matrices". Preprint. Submitted to journal.

\bibitem{BH95} A. Baliga and K.J. Horadam, "Cocyclic Hadamard matrices over $\Z _t \times \Z _2^2$, {\em Australas. J. Combin.}, vol. 11, pp. 123--134, 1995.

\bibitem{Gue11}
M. B. G\"uemes, ``Cocyclic Hadamard matrices over $\Z _t \times
\Z_2^2$. Description, classification and search. PhD thesis (English translation), University of Seville, 2011.

\bibitem{Hor07} K.J. Horadam, {\em Hadamard matrices and their applications}. Princeton: Princeton University Press,  2007.


\bibitem{HdL93} K.J. Horadam and W. de Launey, "Cocyclic development of designs", {\em J. Algebraic Combin.}, vol. 2, no. 3, pp. 267--290, 1993. Erratum, {\em J. Algebraic Combin.}, vol. 3, pp. 129, 1994.

\bibitem{HdL95} K.J. Horadam and W. de Launey, "Generation of cocyclic Hadamard matrices", in: {\em Proc. Computational Algebra and Number Theory}, Sydney, 1992, in {\em Math. Appl.}, vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 279--290.





\bibitem{Kou} C. Koukouvinos, web page, http://www.math.ntua.gr/~ckoukouv/



%\bibitem{HdL93} K.J. Horadam and W. de Launey, "Cocyclic development of designs", {\em J. Algebraic Combin.}, vol. 2, no. 3, pp. 267--290, 1993. Erratum, {\em J. Algebraic Combin.}, vol. 3, pp. 129, 1994.

%\bibitem{HdL95} K.J. Horadam and W. de Launey, "Generation of cocyclic Hadamard matrices", in: {\em Proc. Computational Algebra and Number Theory}, Sydney, 1992, in {\em Math. Appl.}, vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 279--290.















\end{thebibliography}