text:
\section{Introduction}\label{sec:HP}
A {\it Hadamard matrix} of order $v$ is a $v\times v$ matrix $H$ with entries $\pm 1$ such that $HH^{\top} = vI$, where $I$ is the identity matrix. It can be easily shown that if $H$ is a Hadamard matrix of order $v$, then $v=1, 2$, or $4t$ for some positive integer $t$. A long-standing conjecture in combinatorics states that a Hadamard matrix of order $v$ exists for every $v\equiv 0\,(\mod{4})$.  Despite the work of many researchers, the conjecture is far from being resolved. Currently it is still not known whether the set of orders of Hadamard matrices has positive density. For some sparse infinite subsequences of $\{4t: t=1,2,3,\ldots\}$, it is often possible to construct Hadamard matrices of order $v$ for every $v$ belonging to the subsequences. The most famous examples are the Paley constructions which produce Hadamard matrices of order $q+1$ if $q$ is a prime power congruent to 3 modulo 4, and Hadamard matrices of order $2(q+1)$ if $q$ is a prime power congruent to 1 modulo 4. As further examples, we mention that for prime powers $q\equiv 1\,(\mod{4})$ or $q\equiv 3\,(\mod{8})$, Xia and Liu \cite{XL91, XL96} construct Hadamard matrices of order $4q^2$; for $q\equiv 7\,(\mod{8})$, the first author, Ma and Schmidt \cite{LMS06} construct two possibly infinite families of Hadamard matrices of order $4q^2$. All these constructions are based on cyclotomy of finite fields. The Paley constructions use the nonzero squares of $\F_q$. The constructions by Xia and Liu \cite{XL91, XL96}, and by Leung, Ma, and Schmidt \cite{LMS06} use the $4^{\rm th}$, $8^{\rm th}$ and $(q+1)^{\rm th}$ cyclotomic classes of  $\F_{q^2}$. The main idea behind the constructions of Xia/Liu and Leung/Ma/Schmidt is to use cyclotomic classes of finite fields to construct a difference family with appropriate parameters in an abelian group $G$.

Throughout this paper,  we will use the following notation. Let $(G,+)$ be an additively written finite abelian group and  let $G^\ast:=G\setminus \{0_G\}$. For any subset $D$ in $G$, we define
$D^{(-1)}:=\{-x:x \in D\}$,
$\overline{D}:=G^\ast \setminus D$, and $D^c:=G \setminus D$.
Furthermore, we will identify $D$ with the group ring element $\sum_{x\in D}x\in \Z[G]$ when there is no confusion.

Let $B_i$, $i=1,2,\ldots,\ell$, be $k_i$-subsets of $G$.
The set ${\mathcal B}=\{B_i:i=1,2,\ldots,\ell\}$ is called a
{\it difference family with parameters $(v;k_1,k_2,\ldots,k_\ell;\lambda)$ in $G$} if
the list of differences ``$x-y, x,y\in B_i,x\not=y,i=1,2,\ldots,\ell$" represents every nonzero element of $G$ exactly $\lambda$ times; or equivalently
\[ \sum_{i=1}^{\ell} B_i B_i^{(-1)}= \lambda G+ \Big(\sum_{i=1}^{\ell} k_i-\lambda\Big) \cdot 0_G.\]
Each subset $B_i$ is called
a {\it block} of ${\mathcal B}$.
We now define two special classes of difference families.
A difference family in $G$ with four blocks is said to be of {\it type $H$} if $\sum_{i=1}^4 k_i-|G|=\lambda$; and  of
{\it type $H_4^\ast$}
if  $\sum_{i=1}^4 k_i-(|G|+1)=\lambda$.

It is well known that if there is a difference family of type $H$ in $G$, then we obtain a Hadamard matrix of order $4|G|$ by plugging the group invariant
$(-1,1)$ matrices obtained from its blocks into
the Goethals-Seidel array~\cite{GS67}. In the literature, difference families of type $H$ have been extensively studied~\cite{LMS06,XL91,XL95,XL96,XX94,XX99,XXSW05}.

On the other hand, from a difference family of type $H_4^\ast$ in a finite abelian group $G$,  we obtain a Hadamard matrix of order $4(|G|+1)$ by plugging the the group invariant $(-1,1)$ matrices obtained from its blocks into
the Wallis-Whiteman array~\cite[Theorem~4.17]{WSW}.
Indeed difference families of type $H_4^\ast$ are particularly interesting as the orders of the Hadamard matrices obtained from the difference families are no longer of the form $4|G|$, but of the form $4(|G|+1)$. Very recently, the first and second authors \cite{LM18} gave two new constructions of difference families of type $H_4^\ast$ with parameters $(2n;n,n,n,n-1;2n-2)$. Difference families with these parameters were initially considered by Whiteman~\cite{W72}, who obtained one infinite family. Soon afterwards,
Spence~\cite{S75} came up with two new families whose constructions are based on relative difference sets.  On the other hand, the existence of difference families with parameters $(2n;n,n,n,n-1;2n-2)$ in dihedral groups was also studied in \cite{K96,KN02,SY00}. Let us summarize all known constructions of difference families of type $H_4^\ast$ with parameters  $(2n;n,n,n,n-1;2n-2)$.

\begin{theorem}
There exists a difference family of type $H_4^\ast$ with parameters $(2n;n,n,n,$ $n-1;2n-2)$
if $n$ satisfies any of the following conditions:
\begin{itemize}
\item[(1)] {\em \cite{W72,SY00}} $n=q$ and $2q-1$ are both prime powers.
\item[(2)] {\em \cite{S75}} $q=2n+1$ is a prime power for which there exists a nonnegative integer $s$ such that $(q-2^{s+1}-1)/2^{s+1}$ is an odd prime power.
\item[(3)] {\em \cite{S75}} $n=q$ is a prime power such that $q\equiv 1\,(\mod{4})$, and $q-2$ is also a  prime power.
\item[(4)] {\em \cite{LM18}} $n=9^{t_0}q_1^{4t_1}q_2^{4t_2}\cdots q_s^{4t_s}$, where
$p_i$, $i=1,2,\ldots,s$, are prime powers and $t_i$, $i=0,1,\ldots,s$, are  nonnegative integer.
\item[(5)] {\em \cite{LM18}} $n=q^2$ with $q$ a prime power such that $q\equiv 1\,(\mod{4})$.
\end{itemize}
In particular, there exists a Hadamard matrix of order $4(2n+1)$ if $n$ satisfies any of the above conditions.
\end{theorem}
In this paper, we obtain a new series of difference families of type $H_4^*$ with parameters
$(2q^2;q^2,q^2,$ $q^2,q^2-1;2q^2-2)$ where $q$ is a prime power congruent to 3 modulo 8 satisfying some extra condition. The construction uses $8^{\rm th}$ cyclotomic  classes of $\F_{q^2}$ and ``half lines" in $\AG(2,q)$.  In \cite{LMS06,XL91,XL96}, the main idea is to construct difference families of type $H$ in the group $(\F_{q^2},+)$. Our approach here is analogous to that of \cite{LM18}; the main difference here is the usage of Paley type partial difference sets.
The following are our main results.
\begin{theorem}\label{thm:mainth}
Let $q$ be a prime power of the form $q=12c^2+4c+3$ with
$c$ an arbitrary integer, and let $n=q^2$.  Then there exists a difference family with parameters $(2n;n,n,n,n-1;2n-2)$ in $\Z_{2}\times (\F_{q^2},+)$.
\end{theorem}\
By plugging the group invariant $(1,-1)$ matrices obtained from the blocks of the difference family in Theorem~\ref{thm:mainth} into the Wallis-Whiteman array, we immediately obtain the following:
\begin{theorem}\label{thm:mainthm}
Let $q$ be a prime power of the form $q=12c^2+4c+3$ with
$c$ an arbitrary integer, and let $n=q^2$.  Then there exists a
Hadamard matrix of order $4(2n+1)$.
\end{theorem}
We remark that there are $386$ prime powers of the form $q=12c^2+4c+3<10^7$ while there are $166181$ prime powers $q<10^7$ such that $q\equiv 3\,(\mod{8})$. The first $58$ prime powers of  the form $q=12c^2+4c+3<10^5$ are listed below:
\begin{align}
&3, 11, 19, 43, 59, 179, 211, 283, 563, 619, 739, 1163, 1499, 1979, 2083, 2411, 3011,\nonumber\\
&3539, 4259, 4723, 7603, 8011, 8219, 10211, 11411, 12163, 14011, 14563, 14843,\nonumber\\
&17483, 20011, 23059, 25579, 26699, 28619, 29803, 30203, 33923, 36083, 36523, \label{eq:list}\\
& 41539, 49411, 54139, 55219, 55763, 59083, 60779, 63659, 65419, 69011, 70843, \nonumber\\
& 75211, 80363, 81019, 82339, 83003, 88411, 93283.\nonumber
\end{align}

\section{The construction}

We first fix our notation.  Let $q$ be a prime power such that $q\equiv 3\,(\mod{4})$.  Let $\omega$ be a primitive element of $\F_{q^2}$ and
let $0_{\F_{q^2}}$ denote the zero of $\F_{q^2}$. For any fixed positive integer $N$ dividing $q^2-1$, define $C_i^{(N,q^2)}=\omega^i \langle \omega^N\rangle$, $i=0,1,\ldots,N-1$,
called the {\it $N^{\rm th}$ cyclotomic classes} of $\F_{q^2}$. Furthermore, define
\begin{align*}
H_i=&\,C_i^{(2(q+1),q^2)}, \, \, i=0,1,\ldots,2q+1,\\
L_i=&\,C_i^{(q+1,q^2)}, \, \, i=0,1,\ldots,q,\\
S_i=&\,C_i^{(q+1,q^2)}\cup \{0\}, \, \, i=0,1,\ldots,q, \\
 D_i= & C_i^{(4,q^2)}\cup C_{i+1}^{(4,q^2)},  \, \, i=0,1,\ldots,3.
\end{align*}
Note that each $S_i$ is a line through the origin of $\AG(2,q)$; for this reason the $H_i$'s are called half lines \cite{xiang}. In the group ring $\Z[(\F_{q^2},+)]$, we have
\begin{equation}\label{eq:SS}
\mbox{$S_iS_j=\F_{q^2}$ for $i\not=j$ and $S_i^2=qS_i$ for all $i$.}
\end{equation}

\begin{lemma}
\label{lem:pds}
For $i=0,1,2,3$, $D_i$ is a Paley type partial difference set in $(\F_{q^2},+)$. In particular,
\[ D_iD_i^{(-1)}=\frac{q^2-5}{4}D_i+\frac{q^2-1}{4}\overline{D}_i+\frac{q^2-1}{2}\cdot 0_{\F_{q^2}}. \]
\end{lemma}

For a proof of  Lemma~\ref{lem:pds}, we refer the reader to  \cite[p.~216]{peisert}. The strongly regular Cayley graph, ${\rm Cay}(\F_{q^2}, D_0)$, is often called a Peisert graph.

Our objective is to construct difference families with parameters $(2q^2;q^2,q^2-1,q^2,q^2;2q^2-2)$
in $\Z_2\times \F_{q^2}$. So we need to find four blocks $B_0, B_1, B_2, B_3$ with $|B_i|=q^2$, $i=0,2,3$, and $|B_1|=q^2-1$, in $\Z_2\times \F_{q^2}$ such that
\[ \sum_{i=0}^{3} B_i B_i^{(-1)}= (2q^2-2) (\Z_2\times \F_{q^2}) + (2q^2+1) \cdot (0,0_{\F_{q^2}}).\]

To construct the first two blocks, we make use of the Paley type partial difference sets $D_0$ and $D_2$ defined above.  Note that $D_2=\omega^2 D_0$ and $D_2=\overline{D_0}$. In $\Z_{2}\times \F_{q^2}$, we set
\begin{align*}
B_0&\,=(\{0\}\times D_0)\cup (\{1\}\times (\F_{q^2}\setminus D_0)), \\
B_1&\,=(\{0\}\times D_2)\cup (\{1\}\times D_2).
\end{align*}

Then $|B_0|=q^2$ and $|B_1|=q^2-1$.

\begin{proposition}\label{prop:b0b1}
With $B_0, B_1$ defined as above, we have
\begin{equation}\label{eq:firstB}
\sum_{i=0,1}B_iB_i^{(-1)}=
\{0\}\times \Big(
(q^2-2)\F_{q^2}^\ast+(2q^2-1)\cdot 0_{\F_{q^2}}\Big)+\{1\}\times \Big(2D_0-2D_2
+(q^2-1)\F_{q^2}\Big).
\end{equation}
\end{proposition}
\proof
It is clear that
\begin{align}
\sum_{i=0,1}B_iB_i^{(-1)}=&\,
\{0\}\times \Big(2\sum_{i=0,2}D_iD_i^{(-1)}+(q^2-2|D_0|)\F_{q^2}\Big)\nonumber\\
&\, \, +\{1\}\times \Big(-2D_0D_0^{(-1)}+2D_2D_2^{(-1)}
+2|D_0|\F_{q^2}\Big). \label{eq:to1}
\end{align}
By Lemma~\ref{lem:pds}, we have
\begin{align}
\sum_{i=0,2}D_iD_i^{(-1)}=&\,\frac{q^2-5}{4}(D_0+D_2)+\frac{q^2-1}{4}
(\overline{D}_0+\overline{D}_2)+(q^2-1)\cdot 0_{\F_{q^2}}\nonumber\\
=&\, \frac{q^2-3}{2}\F_{q^2}^\ast+(q^2-1)\cdot 0_{\F_{q^2}},   \label{eq:D1}
\end{align}
and
\begin{align}
-D_0D_0^{(-1)}+D_2D_2^{(-1)}=&\,-(\frac{q^2-5}{4}D_0+\frac{q^2-1}{4}D_2)+
(\frac{q^2-5}{4}D_2+\frac{q^2-1}{4}D_0)\nonumber\\
=&\,D_0-D_2. \label{eq:D2}
\end{align}
It is now straight forward to obtain \eqref{eq:firstB} from \eqref{eq:to1}, \eqref{eq:D1} and \eqref{eq:D2}.
\qed
\vspace{0.3cm}

To construct the remaining blocks of the desired difference family, we need difference families of type $H$ in $\F_{q^2}$ that satisfy certain conditions.

\begin{proposition}\label{prop:b0b2}
Suppose  ${\mathcal E}=\{E_i:i=0,1,2,3\}$ is a difference family of type $H$ in $\F_{q^2}$ such that $|E_0|=|E_1|=|E_2|=|E_3|=(q^2-q)/2$ and
\begin{equation}\label{eq:cand_ho}
E_0E_1^{(-1)}+E_1E_0^{(-1)}+E_2E_3^{(-1)}+E_3E_2^{(-1)}=(q-1)^2\F_{q^2}+2D_0
-2D_2.
\end{equation}
Let $B_0, B_1$ be defined as above and set
\begin{align*}
B_2&\,=(\{0\}\times E_0)\cup (\{1\}\times (\F_{q^2}\setminus E_1)), \\
B_3&\,=(\{0\}\times E_2)\cup (\{1\}\times (\F_{q^2}\setminus E_3)).
\end{align*}
Then $\{B_0, B_1, B_2,B_3\}$ is a difference family with parameters $(2q^2;q^2,q^2-1,q^2,q^2;2q^2-2)$
in $\Z_2\times \F_{q^2}$.
\end{proposition}
\proof  First of all, we have $|B_2|=q^2+|E_0|-|E_1|=q^2$ and $|B_3|=q^2+|E_2|-|E_3|=q^2$. In view of \eqref{eq:firstB}, it suffices to show that
\[
\sum_{i=2,3}B_iB_i^{(-1)}=
\{0\}\times (
2q^2\cdot 0_{\F_{q^2}}+q^2 \F_{q^2}^\ast)+\{1\}\times ((q^2-1)\F_{q^2}-2D_0+2D_2).
\]
It is clear that
\begin{align}
\sum_{i=2,3}B_iB_i^{(-1)}=&\,
\{0\}\times \Big(\sum_{i=0}^3E_iE_i^{(-1)}+2(q^2-|E_1|-|E_3|)\F_{q^2}\Big)\nonumber\\
&\, \, +\{1\}\times (-E_0E_1^{(-1)}-E_1E_0^{(-1)}
-E_2E_3^{(-1)}-E_3E_2^{(-1)}
+2(|E_0|+|E_2|)\F_{q^2}). \label{eq:to2}
\end{align}
Since $\{E_i:i=0,1,2,3\}$ is a difference family of type $H$ and
$|E_1|+|E_3|=q^2-q$, we have
\begin{equation}\label{eq:e1}
\sum_{i=0}^3E_iE_i^{(-1)}+2(q^2-|E_1|-|E_3|)\F_{q^2}\\
=2q^2 \cdot 0_{\F_{q^2}}+q^2 \F_{q^2}^\ast.
\end{equation}
On the other hand,
by the assumption~\eqref{eq:cand_ho} and $|E_0|+|E_2|=q^2-q$, we have
\begin{equation}
-E_0E_1^{(-1)}-E_1E_0^{(-1)}
-E_2E_3^{(-1)}-E_3E_2^{(-1)}
+2(|E_0|+|E_2|)\F_{q^2}
=(q^2-1)\F_{q^2}-2D_0+2D_2.\label{eq:e2}
\end{equation}
The proposition now follows from \eqref{eq:to2}, \eqref{eq:e1}, and \eqref{eq:e2}.
\qed

\medskip

To construct difference families of type $H$ in $\F_{q^2}$ satisfying the conditions in Proposition~\ref{prop:b0b2}, it is then natural to consider those constructed in \cite{LMS06}.
\begin{lemma}{\em (\cite[Lemma~4 and Corollary~5]{LMS06})}
\label{lem:scl}
Let $q\equiv 3\,(\mod{4})$ be a prime power and let $e$ be the exact power of $2$ dividing $q+1$.
Let $\alpha<e$ be an odd number and set $\beta=\frac{qe-\alpha(q+1)}{2e}$. Let ${\bf A}\subseteq \{0,1,\ldots,2e-1\}$ and ${\bf B}_0,\ldots,{\bf B}_{e-1}\subseteq \{0,1,\ldots,q\}$ with $|{\bf A}|=\alpha$,
$|{\bf B}_0|=\cdots=|{\bf B}_{e-1}|=\beta$ such that $b\not\equiv a\,(\mod{e})$ for all $a\in {\bf A}$ and $b\in \bigcup_{r=0}^{e-1}{\bf B}_r$. Set \begin{align*}
H&=\bigcup_{i\in {\bf A}}C_{i}^{(2e,q^2)}\\
M_i&=\bigcup_{j\in {\bf B}_i}L_j, \, \, i=0,1,\ldots,e-1\\
{\bf D}_i&=\omega^i(H\cup M_i), \, \, i=0,1,\ldots,e-1.
\end{align*}
Then $|{\bf D}_i|=\frac{q(q-1)}{2}$ for  $i=0,1,\ldots,e-1$, and $\{{\bf D}_i: i=0,1,\ldots,e-1\}$ forms a
difference family in $(\F_{q^2},+)$ with $\lambda=\frac{eq(q-2)}{4}$.
\end{lemma}

We now assume that $q$ is a prime power and $q=8m+3$ for some positive integer $m$. In view of Lemma~\ref{lem:scl}, we need a set ${\bf A}\subseteq \{0,1,\ldots,7\}$ with $|{\bf A}|=3$, and four subsets ${\bf B}_i$, $i=0,1,2,3$, of $\{0,\ldots, q\}$, each of size $m$, satisfying certain conditions.

First, we require $I \cap \{x+4\,(\mod{8}):x \in I\}=\emptyset$. Since $|I|=3$, the condition $I \cap \{x+4\,(\mod{8}):x \in I\}=\emptyset$ simply means that $I$ contains exactly one odd or exactly one even element, say, $y\in I$. (Note that such an $I$ clearly exists, for example, take $I=\{0,1,3\}$; and in this case $y=0$.)  Next, we define two $m$-subsets of $\{0,1,\ldots ,q\}$:
\[ J_1=\{y+2+4i\,(\mod{q+1}):i\in \{0,1,\ldots,m-1\}\} \mbox{ and } \]
\[ J_2=\{y+4i\,(\mod{q+1}):i\in \{0,1,\ldots,m-1\}\}.\]

Now, using the notation in Lemma~\ref{lem:scl}, we set $e=4$, $\alpha=3$ and $\beta=m$.
Let ${\bf A}=I$, ${\bf B}_0={\bf B}_1=J_1$,
\[{\bf B}_2={\bf B}_3=\{y-2+4i\,(\mod{q+1}):i\in \{0,1,\ldots,m-1\}\}.\]
It is then straight forward to check that the conditions in Lemma~\ref{lem:scl} are all satisfied. Therefore we obtain a difference family $\{{\bf D}_i:i=0,1,2,3\}$. However, for our purpose, we need to set $E_0={\bf D}_0, E_1={\bf D}_2, E_2={\bf D}_1$ and $E_3={\bf D}_3$. In terms of $I, J_1, J_2$, we have the following:
\begin{align}
E_0=&\,\Big(\bigcup_{i\in I}C_i^{(8,q^2)}\Big)\cup \Big(\bigcup_{i\in J_1}L_i\Big), \, E_1=\Big(\bigcup_{i\in I}C_{i+2}^{(8,q^2)}\Big)\cup \Big(\bigcup_{i\in J_2}L_i\Big), \label{eq:blocks}\\
E_2=&\,\Big(\bigcup_{i\in I}C_{i+1}^{(8,q^2)}\Big)\cup \Big(\bigcup_{i\in J_1}L_{i+1}\Big), \,
E_3=\Big(\bigcup_{i\in I}C_{i+3}^{(8,q^2)}\Big)\cup \Big(\bigcup_{i\in J_2}L_{i+1}\Big). \nonumber
\end{align}
By Lemma~\ref{lem:scl}, $\{ E_i, i=0,1,2,3\}$ is a difference family of type $H$  in $(\F_{q^2},+)$. Furthermore,
$|E_i|=\frac{q^2-q}{2}$ for $i=0,1,2,3$. It therefore remains to show the following:

\begin{theorem}\label{prop:b0b13}
The $E_i$'s defined in (2.10) satisfy the equation (2.6). In particular, there is a difference family with parameters $(2q^2;q^2,q^2,q^2,q^2-1;2q^2-2)$
in $\Z_2\times (\F_{q^2},+)$.
\end{theorem}

\section{Proof of Theorem ~\ref{prop:b0b13}}

To prove Theorem ~\ref{prop:b0b13}, we need to compute
$E_0E_1^{(-1)}+E_0E_1^{(-1)}+E_2E_3^{(-1)}+E_3E_2^{(-1)}$. As in the case of Lemma 4 in \cite{LMS06}, it will make the computations easier if we write each $E_i$ in a different form (i.e., as a union of $H_i$'s and $L_j$'s).  Recall that $q=8m+3$ is a prime power. We define
\[  I_1=\{x+8i\,(\mod{2(q+1)}) :x\in I, i\in \{0,1,\ldots,2m\}\} \mbox{ and } I_2=I_1+2.\]
Here we use the notation $K+1=\{x+1: x\in K\}$. Note that $|I_1|=|I_2|=3(q+1)/4$. Recall that
\[ J_1=\{y+2+4i\,(\mod{q+1}):i\in \{0,1,\ldots,m-1\}\} \mbox{ and }  J_2=J_1-2.\]
We write
\[ E_0=\sum_{i\in I_1}H_i+\sum_{i\in J_1}L_i \mbox{ and }
E_1=\sum_{i\in I_2}H_i+\sum_{i\in J_2}L_i, \]
\[ E_2=\sum_{i\in I_1+1}H_i+\sum_{i\in J_1+1}L_i \mbox{ and }
E_3=\sum_{i\in I_2+1}H_i+\sum_{i\in J_2+1}L_i. \]
 Observe that the following conditions are satisifed:
\begin{itemize}
\item[(1)] Since $I \cap \{x+4\,(\mod{8}):x \in I\}=\emptyset$, we have $I_i\cap \{h+(q+1)\,(\mod{2(q+1)}):h\in I_i\}=\emptyset$ for $i=1,2$,
\item[(2)] $a\not\equiv b\,(\mod{q+1})$ for all $a\in I_i,b\in J_i$, $i=1,2$,
\item[(3)] $|I_1|+2|J_1|=|I_2|+2|J_2|=q$,
\item[(4)] $J_1\subseteq I_2'\cup J_2$ and  $J_2\subseteq I_1'\cup J_1$,
where $I_j'=\{i\,(\mod{q+1}):i \in I_j\}$ for $j=1,2$.
\end{itemize}

\begin{lemma} \label{lem:basic1}
In the group ring $\Z[(\F_{q^2},+)]$, $E_0E_1^{(-1)}+E_1E_0^{(-1)}=$
\begin{equation}\label{eq:x1x2}
\sum_{i\in I_1}\sum_{j\in I_2}H_iH_j^{(-1)}+
\sum_{i\in I_2}\sum_{j\in I_1}H_iH_j^{(-1)}
+\lambda_1 \cdot 0_{\F_{q^2}}
+\lambda_2\F_{q^2} -|J_2|\sum_{i\in I_1'}S_i-|J_1|\sum_{i\in I_2'}S_i,
\end{equation}
where $\lambda_1=|I_1||J_2|+|I_2||J_1|+2|J_1||J_2|$ and
$\lambda_2=|I_1'||J_2|+|I_2'||J_1|+2|J_1||J_2|-|I_1'\cap J_2|-|I_2'\cap J_1|-2|J_1\cap J_2|$.
\end{lemma}
\proof Note that $L_i^{(-1)}=L_i$. We first expand the expression  $E_0E_1^{(-1)}+E_0E_1^{(-1)}$ and obtain the following:
\[E_0E_1^{(-1)}+E_1E_0^{(-1)}=\sum_{i\in I_1}\sum_{j\in I_2}H_iH_j^{(-1)}+
\sum_{i\in I_2}\sum_{j\in I_1}H_iH_j^{(-1)}+Y \mbox{ where } \]
\[ Y= \sum_{i\in I_1}(H_i+H_i^{(-1)})\sum_{i\in J_2}L_i +\sum_{i\in I_2}(H_i+H_i^{(-1)})\sum_{i\in J_1}L_i + 2\sum_{i\in J_1}L_i\sum_{i\in J_2}L_i.
\]

Note that $S_i=L_i+0_{\F_{q^2}}$. So, we may replace each $L_i$ by $S_i-0_{\F_{q^2}}$ in the above sum and we get
\begin{align}
Y=&\, -|J_2|\sum_{i\in I_1}(H_i+H_i^{(-1)}) -|J_1|\sum_{i\in I_2}(H_i+H_i^{(-1)}) \nonumber\\
 & + \sum_{i\in I_1}\sum_{j\in J_2}S_j(H_i+H_i^{(-1)})+
\sum_{i\in I_2}\sum_{j\in J_1}S_j(H_i+H_i^{(-1)})\nonumber\\
&+2\sum_{i\in J_1}\sum_{j\in J_2}S_iS_j
-2|J_1|\sum_{j\in J_2}S_i-2|J_2|\sum_{j\in J_1}S_i+2|J_1||J_2|\cdot 0_{\F_{q^2}}.
\nonumber
\end{align}

Observe that  $H_i+H_{q+1+i}=S_i- 0_{\F_{q^2}}$ for $i=0,1,\ldots,q$ and (3) holds.
Also note that
\[ \sum_{i\in I_1}\sum_{j\in J_2}S_j(H_i+H_i^{(-1)})= \sum_{i\in I_1'}\sum_{j\in J_2}S_jS_i-|J_2|\sum_{i\in I_1'}S_i \mbox{ and } \]
\[ \sum_{i\in I_2}\sum_{j\in J_1}S_j(H_i+H_i^{(-1)})= \sum_{i\in I_2'}\sum_{j\in J_1}S_jS_i-|J_1|\sum_{i\in I_2'}S_i.\]
We then have
\begin{align}
Y =&\, \lambda_1 \cdot 0_{\F_{q^2}} -|J_2|\sum_{i\in I_1'}S_i -|J_1|\sum_{i\in I_2'}S_i - q\sum_{i\in J_2}S_i -q\sum_{i\in J_1}S_i \nonumber\\
 &
+\sum_{i\in I_1'}\sum_{j\in J_2}S_jS_i+
\sum_{i\in I_2'}\sum_{j\in J_1}S_iS_j
+2\sum_{i\in J_1}\sum_{j\in J_2}S_iS_j.\label{eq:compx1x2}
\end{align}

On the other hand, $S_iS_j=\F_{q^2}$ whenever $i\neq j$. Therefore,
by the conditions~(2) and (4), for distinct $u,v$ in $\{1,2\}$,
\begin{equation}\label{eq:SS2}
\sum_{i\in I_u'}\sum_{j\in J_v}S_jS_i=q\sum_{i\in (I_u'\cap J_v)}S_i+(|I_u'|\cdot |J_v|-|I_u\cap J_v|)\F_{q^2}  \mbox{ and }
\end{equation}
\begin{equation}\label{eq:SS3}
\sum_{i\in J_1}\sum_{j\in J_2}S_iS_j=q\sum_{i\in (J_1\cap J_2)}S_i+(|J_1|\cdot |J_2|-|J_1\cap J_2|)\F_{q^2}.
\end{equation}
\eqref{eq:x1x2} now follows easily from \eqref{eq:compx1x2},  \eqref{eq:SS2} and \eqref{eq:SS3}.
\qed

\medskip

Now, replace $I_i$ with $I_i+1$, and $J_i$ with $J_i+1$ in the argument above and observe that condition (2), (3) and (4) still hold. We immediately get the following:

\begin{lemma} \label{lem:basic2}
In the group ring $\Z[(\F_{q^2},+)]$, $E_2E_3^{(-1)}+E_3E_2^{(-1)}=$
\begin{equation}\label{eq:x1x3}
\sum_{i\in I_1+1}\sum_{j\in I_2+1}H_iH_j^{(-1)}+
\sum_{i\in I_2+1}\sum_{j\in I_1+1}H_iH_j^{(-1)}
+\lambda_1 \cdot 0_{\F_{q^2}}
+\lambda_2\F_{q^2} -|J_2|\sum_{i\in I_1'}S_i-|J_1|\sum_{i\in I_2'}S_i,
\end{equation}
where $\lambda_1=|I_1||J_2|+|I_2||J_1|+2|J_1||J_2|$ and
$\lambda_2=|I_1'||J_2|+|I_2'||J_1|+2|J_1||J_2|-|I_1'\cap J_2|-|I_2'\cap J_1|-2|J_1\cap J_2|$.
\end{lemma}

\medskip

\begin{lemma}\label{lem:eout}
Let $E_i$, $i=0,1,2,3$, be defined as in \eqref{eq:blocks}. Recall that $q=8m+3$. Then, we have
\[ E_0E_1^{(-1)}+E_0E_1^{(-1)}+E_2E_3^{(-1)}+E_3E_2^{(-1)}=\sum_{h=0,1}\sum_{i,j\in I}\Big(C_{i+h}^{(8,q^2)}{C_{j+2+h}^{(8,q^2)}}^{(-1)}+
C_{i+2+h}^{(8,q^2)}{C_{j+h}^{(8,q^2)}}^{(-1)}\Big)+Z \]
where $Z=8m(4m+1)\cdot 0_{\F_{q^2}}+m(28m+5)\F_{q^2}^\ast$.
\end{lemma}
\proof  Applying Lemmas \ref{lem:basic1} and \ref{lem:basic2}, we obtain
\begin{align}
&\,E_0E_1^{(-1)}+E_0E_1^{(-1)}+E_2E_3^{(-1)}+E_3E_2^{(-1)}\nonumber\\
=&\, \sum_{h=0,1}\sum_{i,j\in I}\Big(C_{i+h}^{(8,q^2)}{C_{j+2+h}^{(8,q^2)}}^{(-1)}+
C_{i+2+h}^{(8,q^2)}{C_{j+h}^{(8,q^2)}}^{(-1)}\Big)
+2\lambda_1 0_{\F_{q^2}}
+2\lambda_2\F_{q^2}\label{eq:CC}
\\
&\hspace{1.3cm} -|J_2|\sum_{i\in I_1'}(S_i+S_{i+1})-|J_1|\sum_{i\in I_2'}(S_i+S_{i+1}). \nonumber
\end{align}
Since $|I_1'|=|I_2'|=6m+3$,  we have
\[
\sum_{i\in I_1'}(S_i+S_{i+1})=2(6m+3)\cdot 0_{\F_{q^2}}+\sum_{i\in I}
\Big(C_i^{(4,q^2)}+C_{i+1}^{(4,q^2)}\Big)
\]
and
\[
\sum_{i\in I_2'}(S_i+S_{i+1})=2(6m+3)\cdot 0_{\F_{q^2}}+\sum_{i\in I}
\Big(C_{i+2}^{(4,q^2)}+C_{i+3}^{(4,q^2)}\Big).
\]
Note that $\sum_{j=0}^3 C_{i+j}^{(4,q^2)}=\F_{q^2}^\ast$, $|I|=3$ and $|J_1|=|J_2|=m$. Hence,
\begin{equation}\label{eq:CC1}
-|J_2|\sum_{i\in I_1'}(S_i+S_{i+1})-|J_1|\sum_{i\in I_2'}(S_i+S_{i+1})=
-12m(2m+1)\cdot 0_{\F_{q^2}}-3m\F_{q^2}^\ast.
\end{equation}
Furthermore, it is clear that
\begin{equation}\label{eq:CC2}
\lambda_1=2m(7m+3) \mbox{\, \, and \, }
\lambda_2=2m(7m+2).
\end{equation}
Our lemma now follows from \eqref{eq:CC} with \eqref{eq:CC1} and \eqref{eq:CC2}.
\qed
\vspace{0.3cm}

To finish our proof, we need to evaluate  $C_i^{(8,q^2)}{C_j^{(8,q^2)}}^{(-1)}$. The coefficient $c_x$ of $x\in \F_{q^2}$ in $C_i^{(8,q^2)}{C_j^{(8,q^2)}}^{(-1)}$ is $\big|(C_j^{(8,q^2)}+x)\cap C_i^{(8,q^2)}\big|$. If $x\in C_h^{(8,q^2)}$,
it is clear that $c_x=\big|(C_{j-h}^{(8,q^2)}+1)\cap C_{i-h}^{(8,q^2)}\big|$.
The numbers $(i,j)_{N}=\big|(C_{i}^{(N,q^2)}+1)\cap C_{j}^{(N,q^2)}\big|$, $i,j=0,1,\ldots,N-1$, are called {\it $N^{\rm th}$ cyclotomic numbers}. In our case, $q\equiv 3\,(\mod{8})$ is a prime power.
In view of \cite[Lemma~30]{S67}, we obtain the following:

\begin{proposition} \label{prop:cyclo8}
Let $q\equiv 3\,(\mod{8})$ be a prime power.
Then the cyclotomic numbers $(i,j)_{8}$, $i,j=0,1,\ldots,7$, in $\F_{q^2}$ are determined by Table~\ref{Tab1} and the relations:
\begin{align*}
&64n_1=q^2-15+2q, \, 64n_2=q^2+1-2q-4a,\,
64n_3=q^2+1-6q+8a, \\
&64n_4=q^2+1+18q, \, 64n_5=q^2-7-2q+4a, \,
64n_6=q^2+1+6q+4a+16b, \\
&64n_7=q^2+1+6q+4a-16b, \,
64n_8=q^2-7+2q-8a,
\end{align*}
where $a,b$ are specified by the unique proper representation of
$q^2=a^2+2b^2$ with $a\equiv 1\,(\mod{4})$. Note that there is no restriction on the
sign of $b$.

{\small
\begin{table}[h]
\caption{Cyclotomic numbers of order $8$:
the $(i,j)$-entry is
$(i,j)_8$.}
\label{Tab1}
\vspace{-0.4cm}
$$
\begin{array}{|c||c|c|c|c|c|c|c|c|}
\hline
&0&1&2&3&4&5&6&7\\
\hline
\hline
0&n_1&n_2&n_3&n_2&n_4&n_2&n_3&n_2\\
\hline
1&n_5&n_5&n_6&n_2&n_2&n_2&n_2&n_7\\
\hline
2&n_8&n_2&n_8&n_7&n_3&n_2&n_3&n_6\\
\hline
3&n_5&n_2&n_2&n_5&n_2&n_7&n_6&n_2\\
\hline
4&n_1&n_5&n_8&n_5&n_1&n_5&n_8&n_5\\
\hline
5&n_5&n_2&n_7&n_6&n_2&n_5&n_2&n_2\\
\hline
6&n_8&n_7&n_3&n_2&n_3&n_6&n_8&n_2\\
\hline
7&n_5&n_6&n_2&n_2&n_2&n_2&n_7&n_5\\
\hline
\end{array}
$$
\end{table}}
\end{proposition}

\begin{theorem}\label{thm:b2b3}
Suppose $q^2=a^2+2b^2$ is the unique proper representation with $a\equiv 1\,(\mod{4})$.  Theorem~\ref{prop:b0b13} holds if either of the following conditions is satisfied.
\begin{itemize}
\item [(a)] $I=\{0,2,3\}$ and $3q=a+ 4b+16$.
\item [(b)] $I=\{0,2,7\}$ and  $3q=a- 4b+16$.
\end{itemize}
\end{theorem}
\proof By Lemma~\ref{lem:eout}, it is sufficient to show the following:
\begin{equation}\label{eq:cccc}
U:=\sum_{h=0,1}\sum_{i,j\in I}\Big(C_{i+h}^{(8,q^2)}{C_{j+2+h}^{(8,q^2)}}^{(-1)}+
C_{i+2+h}^{(8,q^2)}{C_{j+h}^{(8,q^2)}}^{(-1)}\Big)=
\frac{q^2-1}{2}\cdot 0_{\F_{q^2}}+\frac{9q^2-17}{16}\F_{q^2}^\ast+2D_0-2D_2.
\end{equation}

We give a proof only in the case where $3q=a+4b+16$. The proof for
the case where $3q=a-4b+16$ is similar.

Define $D=\bigcup_{i\in I}C_i^{(8,q^2)}$, and
let $c_x$ denote the
coefficient  of $x\in \F_{q^2}$ in $U$.
To show that  $c_0=\frac{q^2-1}{2}$, it is sufficient to check number of pairs $(i,j)\in I\times I$ such that
$i\equiv j+2\,(\mod{8})$ and $i+2\equiv j\,(\mod{8})$. Clearly, the solution is $(2,0)$ and $(0,2)$ in each case respectively. Therefore, $c_0=2\times 2 \times \frac{q^2-1}{8}=\frac{q^2-1}{2}$.
 To prove that \eqref{eq:cccc} holds, it is enough to
see that $c_{1}=c_{\omega}=c_{\omega^2}+4=c_{\omega^3}+4$ since
$c_{w^i}=c_{w^{i+4j}}$ for all $i,j$.
On the other hand,
$c_x$ for $x\in \F_{q^2}^\ast$
is given by
\[
c_x=|D\cap (\omega^2 D+x)|+|\omega^2 D\cap (D+x)|+
|\omega D\cap (\omega^3 D+x)|+|\omega^3 D\cap (\omega D+x)|.
\]
Hence, the system of equations $c_{1}=c_{\omega}=c_{\omega^2}+4=c_{\omega^3}+4$ is reformulated as
\begin{align*}
&|D\cap (\omega^2 D+1)|+|\omega^2 D\cap  (D+1)|
+|\omega D\cap (\omega^3 D+1)|+|\omega^3 D\cap  (\omega D+1)|\\
=&\,|\omega^{-1} D\cap (\omega D+1)|+|\omega D\cap  (\omega^{-1} D+1)|+| D\cap (\omega^2 D+1)|+|\omega^2 D\cap  (D+1)|\\
=&\,|\omega^{-2} D\cap (D+1)|+|D\cap  (\omega^{-2} D+1)|+| \omega^{-1}D\cap (\omega D+1)|+|\omega D\cap  (\omega^{-1}D+1)|+4\\
=&\,|\omega^{-3} D\cap (\omega^{-1}D+1)|+|\omega^{-1}D\cap  (\omega^{-3} D+1)|+| \omega^{-2}D\cap (D+1)|+| D\cap  (\omega^{-2}D+1)|+4.
\end{align*}
Noting that $|\omega D\cap (\omega^3 D+1)|+|\omega^3 D\cap (\omega D+1)|=|\omega^{-3} D\cap (\omega^{-1} D+1)|+|\omega^{-1} D\cap (\omega^{-3} D+1)|$,
the equations above are reduced to
\begin{equation}\label{eq:redu1}
|\omega D\cap (\omega^3 D+1)|+|\omega^3 D\cap (\omega D+1)|=
|\omega^{-1} D\cap (\omega D+1)|+|\omega D\cap  (\omega^{-1} D+1)|
\end{equation}
and
\begin{equation}\label{eq:redu2}
|D\cap (\omega^2 D+1)|+|\omega^2 D\cap  (D+1)|=
|\omega^{-2} D\cap  (D+1)|+| D\cap  (\omega^{-2} D+1)|+4.
\end{equation}

Let
\begin{align*}
&N_1=|\omega D\cap (\omega^3 D+1)|+|\omega^3 D\cap  (\omega D+1)|,\quad
N_2=|\omega^{-1} D\cap (\omega D+1)|+|\omega D\cap  (\omega^{-1} D+1)|,\\
&N_3=|D\cap (\omega^2 D+1)|+|\omega^2 D\cap  (D+1)|,\quad
N_4=|\omega^{-2} D\cap  (D+1)|+| D\cap  (\omega^{-2} D+1)|.
\end{align*}
Then, \eqref{eq:redu1} and \eqref{eq:redu2} are rewritten as $N_1=N_2$ and $N_3=N_4+4$, respectively.
From the definition of $I$ and Table~\ref{Tab1} of Proposition~\ref{prop:cyclo8}, we have
\begin{align*}
N_1=&\,(1,3)_8+(1,5)_8+(1,6)_8+(3,3)_8+(3,5)_8+(3,6)_8+(4,3)_8+
(4,5)_8+(4,6)_8\\
&\, +
(3,1)_8+(5,1)_8+(6,1)_8+(3,3)_8+(5,3)_8+(6,3)_8+(3,4)_8+
(5,4)_8+(6,4)_8\\
=&\,8n_2+n_3+4n_5+2n_6+2n_7+n_8, \\
N_2=&\,(7,1)_8+(7,3)_8+(7,4)_8+(1,1)_8+(1,3)_8+(1,4)_8+(2,1)_8+
(2,3)_8+(2,4)_8\\
&\, +
(1,7)_8+(3,7)_8+(4,7)_8+(1,1)_8+(3,1)_8+(4,1)_8+(1,2)_8+
(3,2)_8+(4,2)_8,\\
=&\,
8n_2+n_3+4n_5+2n_6+2n_7+n_8,\\
N_3=&\,(0,2)_8+(0,4)_8+(0,5)_8+(2,2)_8+(2,4)_8+(2,5)_8+(3,2)_8+
(3,4)_8+(3,5)_8\\
&\, +
(2,0)_8+(4,0)_8+(5,0)_8+(2,2)_8+(4,2)_8+(5,2)_8+(2,3)_8+
(4,3)_8+(5,3)_8,\\
=&n_1+4n_2+2n_3+n_4+2n_5+n_6+3n_7+4n_8, \\
N_4=&\,(6,0)_8+(6,2)_8+(6,3)_8+(0,0)_8+(0,2)_8+(0,3)_8+(1,0)_8+
(1,2)_8+(1,3)_8\\
&\, +
(0,6)_8+(2,6)_8+(3,6)_8+(0,0)_8+(2,0)_8+(3,0)_8+(0,1)_8+
(2,1)_8+(3,1)_8\\
=&2n_1+6n_2+4n_3+2n_5+2n_6+2n_8.
\end{align*}
It is clear that $N_1=N_2$.
By the evaluations for $n_1,n_2,\ldots,n_8$ in Proposition~\ref{prop:cyclo8},
we have $N_3=(18q^2+28q-8a-32b-46)/64$ and
$N_4=(18q^2+20q+8a+32b-46)/64$. Hence, $N_3=N_4+4$ if and only if
$3q=a+4b+16$. This shows that \eqref{eq:cccc} holds if $3q=a+4b+16$.
\qed
\vspace{0.3cm}

It is not difficult to see that the condition $q^2=a^2+2b^2$ with $3q=a\pm 4b+16$ and $a\equiv 1\,(\mod{4})$ is equivalent to that $q$ has the form $q=12c^2+4c+3$ with $c$ an arbitrary integer; in this case,  $a=4c^2+12c+1$ and $b=\pm (8c^2-2)$.
Hence, by Theorem~\ref{thm:b2b3} and Proposition~\ref{prop:b0b2},  Theorem~\ref{thm:mainth} now follows.

To see whether we have constructed an infinite family of Hadamard matrices in Theorem~\ref{thm:mainthm}, a natural question arises: are there infinitely many prime powers $q$ of the form $q=12c^2+4c+3$ with $c$ an integer? We believe that there are infinitely many primes of the form $12c^2+4c+3$ with $c$ an integer. But this is probably very difficult to prove. On the other hand, we conjecture that there are no proper prime powers $q$ of the form $q=12c^2+4c+3$ ($c$ is an integer). That is, we conjecture that there are no solutions to the equation
$$12c^2+4c+3=p^{\alpha},\; \alpha>1,$$
where $c$ is an integer, and $p$ is a prime. Some evidence is given in Introduction, namely all $58$ prime powers listed in \eqref{eq:list} are actually primes.

\vspace{0.3cm}

redpajama_set_name: RedPajamaArXiv


text:
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HackensackUMC, a nonprofit teaching and research hospital located in Bergen County, NJ, is the largest provider of inpatient and outpatient services in the state. It is the flagship hospital of the Hackensack University Health Network, one of the largest health networks in the state comprised of 1,140 beds, nearly 10,000 employees and 2,200 credentialed physicians.HackensackUMC was listed as the number one hospital in New Jersey and one of the top four New York metro area hospitals by the U.S. News & World Report, and has received nine national rankings in: Cancer; Cardiology & Heart Surgery; Ear, Nose & Throat; Gastroenterology; Geriatrics; Neurology & Neurosurgery; Orthopedics; Urology; and the Joseph M. Sanzari Children's Hospital ranked as one of the Top 50 Best Children's Hospitals for Neurology and Neurosurgery in the 2013-14 Best Children's Hospitals list. HackensackUMC is among Healthgrades® America's Best 100 Hospitals in 10 different areas – more than any other hospital in the nation, is one of Healthgrades America's 50 Best Hospitals™ for seven years in a row, and received the Healthgrades Distinguished Hospital Award for Clinical Excellence™ 11 years in a row. The medical center has also been named a Leapfrog Top Hospital, and received 19 Gold Seals of Approval™ by the Joint Commission – more than any other hospital in the country. It was the first hospital in New Jersey and second in the nation to become a Magnet® recognized hospital for nursing excellence. HackensackUMC is the Hometown Hospital of the New York Giants and the New York Red Bulls, and remains committed to its community through fundraising and community events. To learn more, visit: www.HackensackUMC.org.


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The Visigoths (; ) were an early Germanic people who, along with the Ostrogoths, constituted the two major political entities of the Goths within the Roman Empire in late antiquity, or what is known as the Migration Period. The Visigoths emerged from earlier Gothic groups, including a large group of Thervingi, who had moved into the Roman Empire beginning in 376 and had played a major role in defeating the Romans at the Battle of Adrianople in 378. Relations between the Romans and the Visigoths varied, with the two groups making treaties when convenient, and warring with one another when not. Under their first leader, Alaric I, the Visigoths invaded Italy and sacked Rome in August 410. Afterwards, they began settling down, first in southern Gaul and eventually in Hispania, where they founded the Visigothic Kingdom and maintained a presence from the 5th to the 8th centuries AD.

The Visigoths first settled in southern Gaul as foederati to the Romans, a relationship that was established in 418. However, they soon fell out with their Roman hosts (for reasons that are now obscure) and established their own kingdom with its capital at Toulouse. They next extended their authority into Hispania at the expense of the Suebi and Vandals.

During their governance of Hispania, the Visigoths built several churches that survived, and left many artifacts which have been discovered in increasing numbers by archaeologists in recent years. The Treasure of Guarrazar of votive crowns and crosses are the most spectacular. In 507, however, their rule in Gaul was ended by the Franks under Clovis I, who defeated them in the Battle of Vouillé. After that, the Visigoth kingdom was limited to Hispania, and they never again held territory north of the Pyrenees other than Septimania, where an elite group of Visigoths came to dominate governance, particularly in the Byzantine province of Spania and the Kingdom of the Suebi.

In or around 589, the Visigoths under Reccared I converted from Arianism to Nicene Christianity, gradually adopting the culture of their Hispano-Roman subjects. Their legal code, the Visigothic Code (completed in 654), abolished the longstanding practice of applying different laws for Romans and Visigoths. Once legal distinctions were no longer being made between Romani and Gothi, they became known collectively as Hispani.

In the century that followed, the region was dominated by the Councils of Toledo and the episcopacy, and little else is known about the Visigoths' history. In 711, an invading force of Arabs and Berbers defeated the Visigoths in the Battle of Guadalete. The Visigoth king, Roderic, and many members of their governing elite were killed, and their kingdom rapidly collapsed.  This was followed by the subsequent formation of the Kingdom of Asturias in northern Spain and the beginning of the Reconquista by Christian troops under Pelagius.

They founded the only new cities in western Europe from the fall of the Western half of the Roman Empire until the rise of the Carolingian dynasty. Many Visigothic names are still in use in modern Spanish and Portuguese languages. Their most notable legacy, however, was the Visigothic Code, which served, among other things, as the basis for court procedure in most of Christian Iberia until the Late Middle Ages, centuries after the demise of the kingdom.

Nomenclature: Vesi, Tervingi, Visigoths

The Visigoths were never called Visigoths, only Goths, until Cassiodorus used the term, when referring to their loss against Clovis I in 507. Cassiodorus apparently invented the term based on the model of the "Ostrogoths", but using the older name of the Vesi, one of the tribal names which the fifth-century poet Sidonius Apollinaris had already used when referring to the Visigoths. The first part of the Ostrogoth name is related to the word "east", and Jordanes, the medieval writer, later clearly contrasted them in his Getica, stating that "Visigoths were the Goths of the western country." According to Wolfram, Cassiodorus created this east–west understanding of the Goths, which was a simplification and literary device, while political realities were more complex. Cassiodorus used the term "Goths" to refer only to the Ostrogoths, whom he served, and reserved the geographic reference "Visigoths" for the Gallo-Spanish Goths. The term "Visigoths" was later used by the Visigoths themselves in their communications with the Byzantine Empire, and was still in use in the 7th century.

Two older tribal names from outside the Roman empire are associated with Visigoths who formed within the empire. The first references to any Gothic tribes by Roman and Greek authors were in the third century, notably including the Thervingi, who were once referred to as Goths by Ammianus Marcellinus. Much less is known of the "Vesi" or "Visi", from whom the term "Visigoth" was derived. Before Sidonius Apollinaris, the Vesi were first mentioned in the Notitia Dignitatum, a late-4th- or early-5th-century list of Roman military forces. This list also contains the last mention of the "Thervingi" in a classical source.

Although he did not refer to the Vesi, Tervingi or Greuthungi, Jordanes identified the Visigothic kings from Alaric I to Alaric II as the successors of the fourth-century Tervingian king Athanaric, and the Ostrogoth kings from Theoderic the Great to Theodahad as the heirs of the Greuthungi king Ermanaric. Based on this, many scholars have traditionally treated the terms "Vesi" and "Tervingi" as referring to one distinct tribe, while the terms "Ostrogothi" and "Greuthungi" were used to refer to another.

Wolfram, who still recently defends the equation of Vesi with the Tervingi, argues that while primary sources occasionally list all four names (as in, for example, Gruthungi, Austrogothi, Tervingi, Visi), whenever they mention two different tribes, they always refer either to "the Vesi and the Ostrogothi" or to "the Tervingi and the Greuthungi", and they never pair them up in any other combination. In addition, Wolfram interprets the Notitia Dignitatum as equating the Vesi with the Tervingi in a reference to the years 388–391. On the other hand, another recent interpretation of the Notitia is that the two names, Vesi and Tervingi, are found in different places in the list, "a clear indication that we are dealing with two different army units, which must also presumably mean that they are, after all, perceived as two different peoples". Peter Heather has written that Wolfram's position is "entirely arguable, but so is the opposite".

Wolfram believes that "Vesi" and "Ostrogothi" were terms each tribe used to boastfully describe itself and argues that "Tervingi" and "Greuthungi" were geographical identifiers each tribe used to describe the other. This would explain why the latter terms dropped out of use shortly after 400, when the Goths were displaced by the Hunnic invasions. Wolfram believes that the people Zosimus describes were those Tervingi who had remained behind after the Hunnic conquest. For the most part, all of the terms discriminating between different Gothic tribes gradually disappeared after they moved into the Roman Empire.

Many recent scholars, such as Peter Heather, have concluded that Visigothic group identity emerged only within the Roman Empire. Roger Collins also believes that the Visigothic identity emerged from the Gothic War of 376–382 when a collection of Tervingi, Greuthungi and other "barbarian" contingents banded together in multiethnic foederati (Wolfram's "federate armies") under Alaric I in the eastern Balkans, since they had become a multi ethnic group and could no longer claim to be exclusively Tervingian.

Other names for other Gothic divisions abounded. In 469, the Visigoths were called the "Alaric Goths". The Frankish Table of Nations, probably of Byzantine or Italian origin, referred to one of the two peoples as the Walagothi, meaning "Roman Goths" (from Germanic *walhaz, foreign). This probably refers to the Romanized Visigoths after their entry into Spain. Landolfus Sagax, writing in the 10th or 11th century, calls the Visigoths the Hypogothi.

Etymology of Tervingi and Vesi/Visigothi
The name Tervingi may mean "forest people", with the first part of the name related to Gothic triu, and English "tree". This is supported by evidence that geographic descriptors were commonly used to distinguish people living north of the Black Sea both before and after Gothic settlement there, by evidence of forest-related names among the Tervingi, and by the lack of evidence for an earlier date for the name pair Tervingi–Greuthungi than the late third century. That the name Tervingi has pre-Pontic, possibly Scandinavian, origins still has support today.

The Visigoths are called Wesi or Wisi by Trebellius Pollio, Claudian and Sidonius Apollinaris. The word is Gothic for "good", implying the "good or worthy people", related to Gothic iusiza "better" and a reflex of Indo-European *wesu "good", akin to Welsh gwiw "excellent", Greek eus "good", Sanskrit vásu-ş "id.". Jordanes relates the tribe's name to a river, though this is probably a folk etymology or legend like his similar story about the Greuthung name.

History

Early origins
The Visigoths emerged from the Gothic tribes, probably a derivative name for the Gutones, a people believed to have their origins in Scandinavia and who migrated southeastwards into eastern Europe. Such understanding of their origins is largely the result of Gothic traditions and their true genesis as a people is as obscure as that of the Franks and Alamanni. The Visigoths spoke an eastern Germanic language that was distinct by the 4th century. Eventually the Gothic language died as a result of contact with other European people during the Middle Ages.

Long struggles between the neighboring Vandili and Lugii people with the Goths may have contributed to their earlier exodus into mainland Europe. The vast majority of them settled between the Oder and Vistula rivers until overpopulation (according to Gothic legends or tribal sagas) forced them to move south and east, where they settled just north of the Black Sea. However, this legend is not supported by archaeological evidence so its validity is disputable. Historian Malcolm Todd contends that while this large en masse migration is possible, the movement of Gothic peoples south-east was probably the result of warrior bands moving closer to the wealth of Ukraine and the cities of the Black Sea coast. Perhaps what is most notable about the Gothic people in this regard was that by the middle of the third century AD, they were "the most formidable military power beyond the lower Danube frontier".

Contact with Rome
Throughout the 3rd and 4th centuries there were numerous conflicts and exchanges of varying types between the Goths and their neighbors. After the Romans withdrew from the territory of Dacia, the local population was subjected to constant invasions by the migratory tribes, among the first being the Goths. In 238, the Goths invaded across the Danube into the Roman province of Moesia, pillaging and exacting payment through hostage taking. During the war with the Persians that year, Goths also appeared in the Roman armies of Gordian III. When subsidies to the Goths were stopped, the Goths organized and in 250 joined a major barbarian invasion led by the Germanic king, Kniva. Success on the battlefield against the Romans inspired additional invasions into the northern Balkans and deeper into Anatolia. Starting in approximately 255, the Goths added a new dimension to their attacks by taking to the sea and invading harbors which brought them into conflict with the Greeks as well. When the city of Pityus fell to the Goths in 256, the Goths were further emboldened. Sometime between 266 and 267, the Goths raided Greece but when they attempted to move into the Bosporus straits to attack Byzantium, they were repulsed. Along with other Germanic tribes, they attacked further into Anatolia, assaulting Crete and Cyprus on the way; shortly thereafter, they pillaged Troy and the temple of Artemis at Ephesus. Throughout the reign of emperor Constantine the Great, the Visigoths continued to conduct raids on Roman territory south of the Danube River. By 332, relations between the Goths and Romans were stabilized by a treaty but this was not to last.

War with Rome (376–382)

The Goths remained in Dacia until 376, when one of their leaders, Fritigern, appealed to the Eastern Roman Emperor Valens to be allowed to settle with his people on the south bank of the Danube. Here, they hoped to find refuge from the Huns. Valens permitted this, as he saw in them "a splendid recruiting ground for his army". However, a famine broke out and Rome was unwilling to supply them with either the food they were promised or the land. Generally, the Goths were abused by the Romans, who began forcing the now starving Goths to trade away their children so as to stave off starvation. Open revolt ensued, leading to 6 years of plundering throughout the Balkans, the death of a Roman Emperor and a disastrous defeat of the Roman army.

The Battle of Adrianople in 378 was the decisive moment of the war. The Roman forces were slaughtered and the Emperor Valens was killed during the fighting. Precisely how Valens fell remains uncertain but Gothic legend tells of how the emperor was taken to a farmhouse, which was set on fire above his head, a tale made more popular by its symbolic representation of a heretical emperor receiving hell's torment. Many of Rome's leading officers and some of their most elite fighting men died during the battle which struck a major blow to Roman prestige and the Empire's military capabilities. Adrianople shocked the Roman world and eventually forced the Romans to negotiate with and settle the tribe within the empire's boundaries, a development with far-reaching consequences for the eventual fall of Rome. Fourth-century Roman soldier and historian Ammianus Marcellinus ended his chronology of Roman history with this battle.

Despite the severe consequences for Rome, Adrianople was not nearly as productive overall for the Visigoths and their gains were short-lived. Still confined to a small and relatively impoverished province of the Empire, another Roman army was being gathered against them, an army which also had amid its ranks other disaffected Goths. Intense campaigns against the Visigoths followed their victory at Adrianople for upwards of three years. Approach routes across the Danube provinces were effectively sealed off by concerted Roman efforts, and while there was no decisive victory to claim, it was essentially a Roman triumph ending in a treaty in 382. The treaty struck with the Goths was to be the first foedus on imperial Roman soil. It required these semi-autonomous Germanic tribes to raise troops for the Roman army in exchange for arable land and freedom from Roman legal structures within the Empire.

Reign of Alaric I

The new emperor, Theodosius I, made peace with the rebels, and this peace held essentially unbroken until Theodosius died in 395. In that year, the Visigoths' most famous king, Alaric I, made a bid for the throne, but controversy and intrigue erupted between the East and West, as General Stilicho tried to maintain his position in the empire. Theodosius was succeeded by his incompetent sons: Arcadius in the east and Honorius in the west. In 397, Alaric was named military commander of the eastern Illyrian prefecture by Arcadius.

Over the next 15 years, an uneasy peace was broken by occasional conflicts between Alaric and the powerful Germanic generals who commanded the Roman armies in the east and west, wielding the real power of the empire. Finally, after the western general Stilicho was executed by Honorius in 408 and the Roman legions massacred the families of thousands of barbarian soldiers who were trying to assimilate into the Roman empire, Alaric decided to march on Rome. After two defeats in Northern Italy and a siege of Rome ended by a negotiated pay-off, Alaric was cheated by another Roman faction. He resolved to cut the city off by capturing its port. On August 24, 410, however, Alaric's troops entered Rome through the Salarian Gate, and sacked the city. However, Rome, while still the official capital, was no longer the de facto seat of the government of the Western Roman Empire. From the late 370s up to 402, Milan was the seat of government, but after the siege of Milan the Imperial Court moved to Ravenna in 402. Honorius visited Rome often, and after his death in 423 the emperors resided mostly there. Rome's fall severely shook the Empire's confidence, especially in the West. Loaded with booty, Alaric and the Visigoths extracted as much as they could with the intention of leaving Italy from Basilicata to northern Africa. Alaric died before the disembarkation and was buried supposedly near the ruins of Croton. He was succeeded by his wife's brother.

Visigothic Kingdom

The Visigothic Kingdom was a Western European power in the 5th to 8th centuries, created first in Gaul, when the Romans lost their control of the western half of their empire and then in Hispania until 711. For a brief period, the Visigoths controlled the strongest kingdom in Western Europe. In response to the invasion of Roman Hispania of 409 by the Vandals, Alans, and Suebi, Honorius, the emperor in the West, enlisted the aid of the Visigoths to regain control of the territory. From 408 to 410 the Visigoths caused so much damage to Rome and the immediate periphery that nearly a decade later, the provinces in and around the city were only able to contribute one-seventh of their previous tax shares.

In 418, Honorius rewarded his Visigothic federates by giving them land in Gallia Aquitania on which to settle after they had attacked the four tribes—Suebi, Asding and Siling Vandals, as well as Alans—who had crossed the Rhine near Mogontiacum (modern Mainz) the last day of 406 and eventually were invited into Spain by a Roman usurper in the autumn of 409 (the latter two tribes were devastated). This was probably done under hospitalitas, the rules for billeting army soldiers. The settlement formed the nucleus of the future Visigothic kingdom that would eventually expand across the Pyrenees and onto the Iberian peninsula. That Visigothic settlement proved paramount to Europe's future as had it not been for the Visigothic warriors who fought side by side with the Roman troops under general Flavius Aetius, it is perhaps possible that Attila would have seized control of Gaul, rather than the Romans being able to retain dominance.

The Visigoths' second great king, Euric, unified the various quarreling factions among the Visigoths and, in 475, concluded the peace treaty with the emperor Julius Nepos. In the treaty the emperor was called a friend (amicus) to the Visigoths, while requiring them to address him as lord (dominus). Though the emperor did not legally recognize Gothic sovereignty, according to some views under this treaty the Visigothic kingdom became an independent kingdom. Between 471 and 476, Euric captured most of southern Gaul. According to historian J. B. Bury, Euric was probably the "greatest of the Visigothic kings" for he managed to secure territorial gains denied to his predecessors and even acquired access to the Mediterranean Sea. At his death, the Visigoths were the most powerful of the successor states to the Western Roman Empire and were at the very height of their power. Not only had Euric secured significant territory, he and his son, Alaric II, who succeeded him, adopted Roman administrative and bureaucratic governance, including Rome's tax gathering policies and legal codes.

At this point, the Visigoths were also the dominant power in the Iberian Peninsula, quickly crushing the Alans and forcing the Vandals into north Africa. By 500, the Visigothic Kingdom, centred at Toulouse, controlled Aquitania and Gallia Narbonensis and most of Hispania with the exception of the Kingdom of the Suebi in the northwest and small areas controlled by the Basques and Cantabrians. Any survey of western Europe taken during this moment would have led one to conclude that the very future of Europe itself "depended on the Visigoths". However, in 507, the Franks under Clovis I defeated the Visigoths in the Battle of Vouillé and wrested control of Aquitaine. King Alaric II was killed in battle. French national myths romanticize this moment as the time when a previously divided Gaul morphed into the united kingdom of Francia under Clovis.

Visigothic power throughout Gaul was not lost in its entirety due to the support from the powerful Ostrogothic king in Italy, Theodoric the Great, whose forces pushed Clovis I and his armies out of Visigothic territories. Theodoric the Great's assistance was not some expression of ethnic altruism, but formed part of his plan to extend his power across Spain and its associated lands.

After Alaric II's death, Visigothic nobles spirited his heir, the child-king Amalaric, first to Narbonne, which was the last Gothic outpost in Gaul, and further across the Pyrenees into Hispania. The center of Visigothic rule shifted first to Barcelona, then inland and south to Toledo. From 511 to 526, the Visigoths were ruled by Theoderic the Great of the Ostrogoths as de jure regent for the young Amalaric. Theodoric's death in 526, however, enabled the Visigoths to restore their royal line and re-partition the Visigothic kingdom through Amalaric, who incidentally, was more than just Alaric II's son; he was also the grandson of Theodoric the Great through his daughter Theodegotho. Amalaric reigned independently for five years. Following Amalaric's assassination in 531, another Ostrogothic ruler, Theudis took his place. For the next seventeen years, Theudis held the Visigothic throne.

Sometime in 549, the Visigoth Athanagild sought military assistance from Justinian I and while this aide helped Athanagild win his wars, the Romans had much more in mind. Granada and southernmost Baetica were lost to representatives of the Byzantine Empire (to form the province of Spania) who had been invited in to help settle this Visigothic dynastic struggle, but who stayed on, as a hoped-for spearhead to a "Reconquest" of the far west envisaged by emperor Justinian I. Imperial Roman armies took advantage of Visigothic rivalries and established a government at Córdoba.

The last Arian Visigothic king, Liuvigild, conquered most of the northern regions (Cantabria) in 574, the Suevic kingdom in 584, and regained part of the southern areas lost to the Byzantines, which King Suintila recovered in 624. Suintila reigned until 631. Only one historical source was written between the years 625 through 711, which comes from Julian of Toledo and only deals with the years 672 and 673. Wamba was the king of the Visigoths from 672 to 680. During his reign, the Visigothic kingdom encompassed all of Hispania and part of southern Gaul known as Septimania. Wamba was succeeded by King Ervig, whose rule lasted until 687. Collins observes that "Ervig proclaimed Egica as his chosen successor" on 14 November 687. In 700, Egica's son Wittiza followed him on the throne according to the Chronica Regum Visigothorum.

The kingdom survived until 711, when King Roderic (Rodrigo) was killed while opposing an invasion from the south by the Umayyad Caliphate in the Battle of Guadalete. This marked the beginning of the Umayyad conquest of Hispania, when most of the Iberian Peninsula came under Islamic rule in the early 8th century.

A Visigothic nobleman, Pelayo, is credited with beginning the Christian Reconquista of Iberia in 718, when he defeated the Umayyad forces in the Battle of Covadonga and established the Kingdom of Asturias in the northern part of the peninsula. According to Joseph F. O'Callaghan, the remnants of the Hispano-Gothic aristocracy still played an important role in the society of Hispania. At the end of Visigothic rule, the assimilation of Hispano-Romans and Visigoths was occurring at a fast pace. Their nobility had begun to think of themselves as constituting one people, the gens Gothorum or the Hispani. An unknown number of them fled and took refuge in Asturias or Septimania. In Asturias they supported Pelagius's uprising, and joining with the indigenous leaders, formed a new aristocracy. The population of the mountain region consisted of native Astures, Galicians, Cantabri, Basques and other groups unassimilated into Hispano-Gothic society. Other Visigoths who refused to adopt the Muslim faith or live under their rule, fled north to the kingdom of the Franks, and Visigoths played key roles in the empire of Charlemagne a few generations later. In the early years of the Emirate of Córdoba, a group of Visigoths who remained under Muslim dominance constituted the personal bodyguard of the Emir, al-Haras.

During their long reign in Spain, the Visigoths were responsible for the only new cities founded in Western Europe between the 5th and 8th centuries. It is certain (through contemporary Spanish accounts) that they founded four: Reccopolis, Victoriacum (modern Vitoria-Gasteiz, though perhaps Iruña-Veleia), Luceo and Olite. There is also a possible 5th city ascribed to them by a later Arabic source: Baiyara (perhaps modern Montoro). All of these cities were founded for military purposes and three of them in celebration of victory. Despite the fact that the Visigoths reigned in Spain for upwards of 250 years, there are few remnants of the Gothic language borrowed into Spanish.
The Visigoths as heirs of the Roman empire lost their language and intermarried with the Hispano-Roman population of Spain.

Genetics

A genetic study published in Science in March 2019 examined the remains of eight Visigoths buried at Pla de l'Horta in the 6th century. These individuals displayed genetic links to northern and central Europe.

Culture

Law
The Visigothic Code of Law (Latin: Forum Iudicum), also called Liber Iudiciorum ( English: Book of the Judges) and Lex Visigothorum (English: Law of the Visigoths), is a set of laws first promulgated by king Chindasuinth (642–653 AD) which had been part of aristocratic oral tradition, was set in writing in the year 654 and survives in two separate codices preserved at el Escorial (Spain). It goes into more detail than a modern constitution commonly does and reveals a great deal about Visigothic social structure. The code abolished the old tradition of having different laws for Romans (leges romanae) and Visigoths (leges barbarorum), and under which all the subjects of the Visigothic kingdom ceased being romani and gothi and instead became hispani. All the kingdom's subjects were under the same jurisdiction, which eliminated social and legal differences and facilitated greater assimilation of the various population groups. The Visigothic Code marks the transition from Roman law to Germanic law.

One of the greatest contributions of the Visigoths to family law was their protection of the property rights of married women, which was continued by Spanish law and ultimately evolved into the community property system now in force throughout the majority of western Europe.

Religion
Before the Middle Ages, the Visigoths, as well as other Germanic peoples, followed what is now referred to as Germanic paganism. While the Germanic peoples were slowly converted to Christianity by varying means, many elements of the pre-Christian culture and indigenous beliefs remained firmly in place after the conversion process, particularly in the more rural and distant regions.

The Visigoths, Ostrogoths and Vandals were Christianized while they were still outside the bounds of the Roman Empire; however, they converted to Arianism rather than to the Nicene version (Trinitarianism) followed by most Romans, who considered them heretics. There was a religious gulf between the Visigoths, who had for a long time adhered to Arianism, and their Catholic subjects in Hispania. There were also deep sectarian splits among the Catholic population of the peninsula which contributed to the toleration of the Arian Visigoths on the peninsula. The Visigoths scorned to interfere among Catholics but were interested in decorum and public order. King Liuvigild (568–586), attempted to restore political unity between the Visigothic-Arian elite and the Hispano-Roman Nicene Catholic population through a doctrinal settlement of compromise on matters of faith, but this failed. Sources indicate that the Iberian Visigoths maintained their Christian Arianism, especially the Visigothic elite until the end of Liuvigild's reign. When Reccared I converted to Catholicism, he sought to unify the kingdom under a single faith.

While the Visigoths retained their Arian faith, the Jews were well tolerated. Previous Roman and Byzantine law determined their status, and it already sharply discriminated against them, but royal jurisdiction was in any case quite limited: local lords and populations related to Jews as they saw fit. We read of rabbis being asked by non-Jews to bless their fields, for example. Historian Jane Gerber relates that some of the Jews "held ranking posts in the government or the army; others were recruited and organized for garrison service; still others continued to hold senatorial rank". In general, then, they were well respected and well treated by the Visigothic kings, that is, until their transition from Arianism to Catholicism. Conversion to Catholicism across Visigothic society reduced much of the friction between the Visigoths and the Hispano-Roman population. However, the Visigothic conversion negatively impacted the Jews, who came under scrutiny for their religious practices.

King Reccared convened the Third Council of Toledo to settle religious disputations related to the religious conversion from Arianism to Catholicism. The discriminatory laws passed at this Council seem not to have been universally enforced, however, as indicated by several more Councils of Toledo that repeated these laws and extended their stringency. These entered canon law and became legal precedents in other parts of Europe as well. The culmination of this process occurred under King Sisibut, who officially decreed a forced Christian conversion upon all Jews residing in Spain. This mandate apparently achieved only partial success: similar decrees were repeated by later kings as central power was consolidated. These laws either prescribed forcible baptism of the Jews or forbade circumcision, Jewish rites, and the observance of the Sabbath and other festivals. Throughout the 7th century the Jews were persecuted for religious reasons, had their property confiscated, were subjected to ruinous taxes, forbidden to trade and, at times, dragged to the baptismal font. Many were obliged to accept Christianity but continued privately to observe the Jewish religion and practices. The decree of 613 set off a century of difficulty for Spanish Jewry, which was only ended by the Muslim conquest.

The political aspects of the imposition of Church power cannot be ignored in these matters. With the conversion of the Visigothic kings to Chalcedonian Christianity, the bishops increased their power, until, at the Fourth Council of Toledo in 633, they selected a king from among the royal family, a practice previously reserved for nobles. This was the same synod that spoke out against those who had been baptized but had relapsed into Judaism. As far as the Visigoths were concerned, the time for religious pluralism "was past". By the end of the 7th century, Catholic conversion made the Visigoths less distinguishable from the indigenous Roman citizens of the Iberian peninsula; when the last Visigothic strongholds fell to the Muslim armies, whose subsequent invasions transformed Spain from the beginning of the 8th century, their Gothic identity faded.

In the eighth through 11th centuries, the muwallad clan of the Banu Qasi claimed descent from the Visigothic Count Cassius.

Architecture

During their governance of Hispania, the Visigoths built several churches in the basilical or cruciform style that survive, including the churches of San Pedro de la Nave in El Campillo, Santa María de Melque in San Martín de Montalbán, Santa Lucía del Trampal in Alcuéscar, Santa Comba in Bande, and Santa María de Lara in Quintanilla de las Viñas. The Visigothic crypt (the Crypt of San Antolín) in the Palencia Cathedral is a Visigothic chapel from the mid-7th century, built during the reign of Wamba to preserve the remains of the martyr Saint Antoninus of Pamiers, a Visigothic-Gallic nobleman brought from Narbonne to Visigothic Hispania in 672 or 673 by Wamba himself. These are the only remains of the Visigothic cathedral of Palencia.

Reccopolis, located near the tiny modern village of Zorita de los Canes in the province of Guadalajara, Castile-La Mancha, Spain, is an archaeological site of one of at least four cities founded in Hispania by the Visigoths. It is the only city in Western Europe to have been founded between the 5th and 8th centuries. The city's construction was ordered by the Visigothic king Liuvigild to honor his son Reccared and to serve as Reccared's seat as co-king in the Visigothic province of Celtiberia, to the west of Carpetania, where the main capital, Toledo, lay.

Goldsmithery

In Spain, an important collection of Visigothic metalwork was found in Guadamur, in the Province of Toledo, known as the Treasure of Guarrazar. This archeological find is composed of twenty-six votive crowns and gold crosses from the royal workshop in Toledo, with signs of Byzantine influence. According to Spanish archaeologists, this treasure represents the high point of Visigothic goldsmithery. The two most important votive crowns are those of Recceswinth and of Suintila, displayed in the National Archaeological Museum of Madrid; both are made of gold, encrusted with sapphires, pearls and other precious stones.
The discoverer of the second lot gave Spanish Queen Elizabeth II some of the pieces that she still had in her possession, including the crown of Suintila, this crown was stolen in 1921 and never recovered. There are several other small crowns and many votive crosses in the treasure.

These findings, along with others from some neighbouring sites and with the archaeological excavation of the Spanish Ministry of Public Works and the Royal Spanish Academy of History (April 1859), formed a group consisting of:

 National Archaeological Museum of Spain: six crowns, five crosses, a pendant and remnants of foil and channels (almost all of gold).
 Royal Palace of Madrid: a crown and a gold cross and a stone engraved with the Annunciation. A crown and other fragments of a tiller with a crystal ball were stolen from the Royal Palace of Madrid in 1921 and its whereabouts are still unknown.
 National Museum of the Middle Ages, Paris: three crowns, two crosses, links and gold pendants.

The aquiliform (eagle-shaped) fibulae that have been discovered in necropolises such as Duratón, Madrona or Castiltierra (cities of Segovia), are an unmistakable example of the Visigothic presence in Spain. These fibulae were used individually or in pairs, as clasps or pins in gold, bronze and glass to join clothes, showing the work of the goldsmiths of Visigothic Hispania.

The Visigothic belt buckles, a symbol of rank and status characteristic of Visigothic women's clothing, are also notable as works of goldsmithery. Some pieces contain exceptional Byzantine-style lapis lazuli inlays and are generally rectangular in shape, with copper alloy, garnets and glass.

See also
Romano-Germanic culture
Thiufa
Goths
Visigothic kingdom
Visigothic art and architecture

References

Notes

Citations

Bibliography

External links

Lex Visigothorum
Visigothic Symposia
National Archaeological Museum-Museo Arqueológico Nacional of Spain

Early Germanic peoples
Germanic tribes
Iron Age peoples of Europe
5th century in sub-Roman Gaul
History of Asturias

redpajama_set_name: RedPajamaWikipedia


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DaVinci Resolve Full Cracked [Final Version]
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DaVinci Resolve Download Cracked + with key [September 2022]
Theres no question that davinci resolve free video editors raw editor is basic, but its many advanced tools, effects, and sharing tools allow it to rival the power of an offline editing suite.
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What is DaVinci Resolve and what is it for
DaVinci Resolves raw editor is easy to use and understand and with over 40 hours of tutorials from the software company, youd be forgiven for thinking its one of the most powerful NLEs available.
Theres just one problem: theres no quick access to tools, like the V-snapper, which you may find invaluable for your projects. If you have recently taken a look at the new DaVinci Resolve 14.2, youll notice that theres been some significant improvements to the Editing workspace. But theres also a massive overhaul to the Project workspace. This means theres been a drastic reduction of tool shortcuts. Theres no F key for selecting your current clip, theres no V key to go back to the previous frame, and theres no X to switch between clips. This is a major problem.
Theres a lot to davinci resolve free video editors interface and youll need a pretty powerful machine to edit in DaVinci Resolve. If you have, a good graphics card, a minimum of 16GB of RAM, and a good CPU, then you should be able to make a lot of progress editing your projects. With an iMac Pro, for example, youll be able to throw up a few of your creations.
The one feature youll absolutely need is some sort of audio sync which is most commonly found in video editing tools. For example, theres a channel option in a lot of programs that does this for you and itll allow you to build a timeline one audio channel at a time. Resolve does this for you and much more.
To be honest, davinci resolve free video editor sounds like an overkill choice for a lot of folks. Sure, if youre really trying to cut back on the amount of time you spend in video editing, youre probably making a good decision to jump straight to a tool like DaVinci Resolve. Blackmagic continued to build its tool for professionals and creators, who are looking to make their jobs easier by streamlining their video processing. As more and more people moved away from the big giant stinking programs, I think their market was just stretched thin to the point where they needed a replacement.
However, it doesnt take a rocket scientist to see that davinci resolve free video editor is really intended for a small army of professionals, who have the time and money to build a tool that does what the other programs do, but does it faster and easier.
Okay, whatever you choose to call it, its the oldest and most robust video editing tool on the market. We'll dive into the good, the bad, and why you should run out and get the software of your choice.
DaVinci Resolve is a powerful color correction application to enhance the look of your videos by correcting, compositing, enhancing, and creating a 3D Depth of Field effect.
Its versatile and completely unique workflow enables you to create productions with a different look in each of its over 40 tools and add the visual effects you need to turn your videos into reality. You could apply graphics, create a 3D Depth of Field, and even create a custom viewer for your next keynote.
The workflow of DaVinci Resolve is completely unique. It works in a similar workflow as Final Cut Pro X, which means you drag and drop clips and sequences into the sequence editor, and use the inspector and timeline to change your artwork, effects and timeline. davinci resolve free video editor is not a stand-alone application, and it doesn't have a timeline. You simply move clips on the timeline, and you can alter the art style in multiple panels including the timeline or inspector.
You can move the timeline wherever you wish by dragging and dropping your clips on the timeline. Dragging and dropping is very important in DaVinci Resolve, and it also allows you to see each and every change instantly without any delay. The next step is to change your clip order in the timeline by dragging them and dropping them in their new order.
Once the timeline is set to the desired order, then you can edit the clips and set the color correction effect or other visual effects if needed. You can also alter the speed of the clip and change the aspect ratio, and you can even change the audio volume if needed. The timeline in davinci resolve free video editor is totally flexible, and you can do anything you want with it.
DaVinci Resolve Studio Features
High Definition HD Editing
Project libraries can be edited using full HD resolution without having to convert the files. The HD resolution is native to the project format without post production conversion of the resolution.
Professional Workflow Modes
The Studio edition of davinci resolve free video editor includes the latest and best functionality for post workflow. Professional modes such as noise reduction, deinterlacing, color grading and more are at your fingertips. In addition, Resolve Studio includes professional modes for Blackmagic Fusion graphics, motion tracking and iClips.
Real Time Color Correction
Color grading can be performed in real-time thanks to the automatic predictions of real-time playback that are based on data from the DaVinci Resolve color engine. The flexible color edit workflow can be performed in real time. Real-time color correction allows for the finest control of every aspect of the image, all within the platform. Color grading, levels, gamma, clarity, color temperature, contrast and more can be adjusted to your satisfaction in one powerful workflow.
Multiple Editors
davinci resolve free video editor is deeply integrated into the Blackmagic Fusion platform. Multiple editors can work on the same project, in real time. These editors include: Editors, Fixers, Trackers, Animators, Titles, Compositors, Colorists, etc. Editors can work independently on a timeline or as a team.
Speed and Performance
The DaVinci Resolve interface is optimized for speed with a smooth scrolling timeline and streamlined controls. The new Resolve Settings panel provides quick access to preferences that can be changed to suit an individual workflow. Resolve Settings can be updated at any time.
DaVinci Resolve Studio 18 features over 100 GPU and CPU accelerated Resolve FX. Resolve effects include blurs, light effects, noise, image restoration, beauty enhancement, stylizing, and more! Version 18 adds even more plugins for depth map generation, surface tracking, fast noise, and despilling. There are even improvements to the beauty effect, edge detection, and lens reflections. Automatically generate a 3D matte of a scene with the depth map FX to grade or add effects to a background or foreground. Track the movement of textured surfaces with the surface tracker effect to apply images to t-shirts, flags, and faces! All Resolve FX effects can be applied and animated in the cut, edit, Fusion, and color pages!
When youre ready to deliver a soundtrack, there are numerous options to meet your needs. To export the finished video with a simple stereo mix, switch to the edit page and use the quick export option in the file menu. For more delivery options, go to the deliver page where youll find a variety of render presets ranging from a custom video clip, to rendering audio only files. Use the Audio Settings to select the audio format, buses and tracks you want to render. davinci resolve free video editor with Fairlight gives you incredibly rich mastering and sound finishing options. You can output everything from mono and stereo to immersive 5.1, 7.1, 22.2 and even the latest 3D audio formats, all from within the same project!
The DaVinci Neural Engine uses state-of-the-art deep neural networks and machine learning, along with artificial intelligence to power features such as facial recognition, object detection, smart reframing, speed warp retiming, and super scale upscaling, auto color, and color matching, and more! In fact, the DaVinci Neural Engine is entirely cross-platform. It uses the latest GPU innovations for AI and deep learning to provide unparalleled performance and quality. The DaVinci Neural Engine provides simple tools to solve complex, repetitive, and time-consuming problems. For example, it enables facial recognition to sort and organize clips into bins based on people in the shot, reframe shots, and more.
This release of Resolve includes a number of new key features that will make your editing and colour grading fun and more efficient. DaVinciResolve Studio is released with all features and content from DCP 2015 on this first release. This update resolves a number of bugs and issues from the previous DCP releases. Some examples of the issuesresolved include:
This release of Resolve has been tested and certified on Windows and Mac operating systems. The new content and features are currently ready to use and the update is free. For thefirst time ever, DaVinci Resolve 18 is also available for Linux and OSX and is fully compatible with Linux and Mac as well. A single davinci resolve free video editor Studio installation can contain all modules, including Resolve, Resolve Studio and Resolve Scratch.
The new version of DaVinci Resolve displays over 50 more channels, from 50 to 1,060, in the new Multicam displays, as well as the ability to upscale almostany existing format to a wider display (up to 8K).
When working on large projects (i.e. most editors, colorists, and sound editors), Resolve has a new mode to help you work more comfortably and efficiently. The 'alternate timeline' support mode showswhere in the timeline you are in real time, allowing you to jump straight to the right section of the timeline.
DaVinci Resolve 18 was developed by Blackmagic Design, the company behind the popular davinci resolve free video editor editing software. Resolve 2018 is the newest version of the editing software, which is also known as Resolve. This version is the third version to be released.
While DaVinci Resolve 18 is a major update, it isn't a brand-new version of the software. The Beta and RC versions were released a few weeks ago, while the first final version was released just five days ago. This version brings davinci resolve free video editor closer to its original promise. Blackmagic Design has finally created a standalone edit, grade, and finish tool for editors. You can now use this powerful tool on its own, without needing any other software.
In DaVinci Resolve, you get the same streamlined workflow that you get in any editing software. Other than that, Resolve comes with lots of features and options that give you complete control over your editing process. Whether you are editing for film or video, live or After Effects, davinci resolve free video editor gives you the best editing tool in the industry. Any video editor can use DaVinci Resolve to give their projects the best editing experience, without having to spend any time learning a new system. Even though davinci resolve free video editor came from Blackmagic Design, it's a top-notch solution.
DaVinci Resolve 18 offers support for the Blackmagic Design interfaces such as URSA Mini, Blackmagic Design Resolve Studio, Blackmagic Design URSA Pro, and Resolve Studio. It also integrates with Apple Final Cut Pro X. Of course, the latest version of DaVinci Resolve 18 is compatible with Blackmagic Design davinci resolve free video editor Studio R14.
One of the most exciting new features in DaVinci Resolve 2018 is the Blackmagic Resolve Cloud. This version of davinci resolve free video editor finally allows you to share your projects with others. It also supports collaboration with Resolve for iPad and Resolve Studio. You can share projects with friends and family using the Blackmagic Cloud.
Other than that, this version of Resolve also allows you to use a proxy workflow. This workflow allows you to create a proxy that is used in DaVinci Resolve.
And there are more reasons to love davinci resolve free video editor than I could possibly list here, so make sure you check out MZed.com/DaVinci-Resolve/Editing-with-DaVinci-Resolve-Part-1-of-2-cf99b8ea9dda#.WhxF6z7qFq">Part 1 of two if youre new to the software and want to get a jumpstart on it.
If you already have an editing platform and want to learn DaVinci Resolve, pick up a copy of MZed City, Volume 1: davinci resolve free video editor or Zane Featherstone: Beyond the Frame.
There are a few main benefits to DaVinci Resolve that can make it easier to get into color grading and make your life easier. First, it is incredibly cheap compared to other color-grading tools like Adobe Premier or FCPX, or even the very expensive Adobe Lavin. A license for its $219 Color Grade Edition is only $50 per user. Second, its powered by Apple and most arent. Third, it has a lot of power and flexibility in its stock features without any complications or licenses. And finally, blackmagic or Im good enough that learning davinci resolve free video editor doesnt even feel like work, and Im excited to get started.
• It doesnt support motion graphics (motion graphics editing in Photoshop is way better than Resolves Motion Graphics panels)
• It doesnt support segmented editing (this is one reason that Adobe Resolve is way better than Adobe Premiere)
• It doesnt support compositing (many people will never use anything more powerful than an average compositing app. I can do most of my compositing in The Foundry Houdini and have had no problems)
• It doesnt have the extensibility of Adobe Resolve. While its cool to be able to call up and control plugins, Adobe Resolve can give you access to 50,000 plugins. DaVinci Resolve will give you access to davinci resolve free video editors own plugins and the odd third-party plugin, but that isnt much more than the feature set of other NLEs. All in all, for the money, I prefer Adobe Resolve. Its far more powerful.
I have assembled a list of links for you to refer to if you want to learn more about DaVinci Resolve. I would strongly suggest subscribing to the Blackmagic Design blog – Ollie has an extensive knowledge of color grading and davinci resolve free video editor, and theres a lot of general information in his articles.
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I'm going to briefly outline what's in the 18 version for quick reference. For a more detailed review, please visit the official DaVinci Resolve Studio review on the Blackmagic website.
Part of Adobe's Creative Cloud suite, davinci resolve free video editor is a professional multi-track editor and colorist. The application functions similar to a final cut editor, with some unique features.
Adobe's newest flagship application, After Effects, seems to be losing customers to the Premiere Pro editing platform, and while Resolve may not be quite as popular as Premiere, it seems to be gaining ground. Resolve now comes in two flavors: Studio and Speed.
The Resolve Studio and Speed Editions are geared towards the entry-level and professional editing market. The difference being, that Studio is free and Speed, while billed at $249, offers a few more features and lower price-point. Neither version offers editing on a live clip basis, which is available with the Premiere Pro editing package. (Only the $499 professional version of Premiere can do this.)
An amazing amount of functionality and you get it all for only $69.99. The product can be upgraded with Media Composer Dynamic Link licenses, which increase functionality and allow you to use apps in Resolve with the greater functionality of Media Composer, for a total of $349.99 per license.
The truth is most of the functionality is built in. Custom processing, real-time transformations, color correction, and a large selection of effects are all preinstalled. Not only that, but some of the most important plugins are preinstalled in a "free" user space. This is where editing in Resolve is done, typically during a final workflow. After editing, some editing tools can be moved to Media Composer for the final rendering into a DNxHD Proxy Plus file. This process allows for easy re-rendering into a new format or another editing system. Because of this, some tools and settings aren't even labeled.
If you need a feature that is available to both user spaces, you will need to "buy" it. For example, you will need to purchase the plugins that make up Full Resolve, a powerful audio editing tool for real time noise reduction. Full Resolve is $449.99 and requires the full version of DaVinci Resolve. This means you will have to pay for the company's support services, which includes such things as monthly training and upgrades. That, and the tax. And the cost of downloading the software. The software has an unofficial upgrade process, but the user space is limited and it may not work with some editing tools.
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davinci resolve free video editor 18 continues to improve on a stable of exciting features. Adobe's newest flagship color post-production software offers new ways to smooth out the rough edges of your footage. Applied to your clips, this method gives you new options for selective color correction. I found that with only a little bit of color adjustment, I could get noticeable results. You can also correct the skin tone in your footage using the GraduatedFilter. With just a few clicks you can get the skin to match the color of your background.
Make it easier to trim your clips in DaVinci Resolve 18. Now you can preview longer clips first, reducing the time it takes to trim and perfect the start and end of your project. In just seconds you can trim an entire clip to your liking. In addition, you can now use your preferred keyboard shortcuts for faster, more effective editing.
davinci resolve free video editor 18 is faster, has more powerful editing tools and is completely new software design. With the new timeline, you can enjoy a simpler editing workflow, and get to know your footage faster.
Videocopilot is an innovative new feature in DaVinci Resolve 18 which allows video to be seamlessly replaced with other assets. You can add any video to your timeline and set the original to scale in or out. The original can be any asset including an image, logo or color, or even a different movie. By inserting the video into the timeline, in seconds you can add a Hollywood look to your story with a few clicks!
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DaVinci Resolve is much more than a video editor. Video editing is a fundamental part of the product, but it's not the only thing. With davinci resolve free video editor, Apple has also created a timeline to edit RAW, HD and 2K images, so I'm going to talk about RAW editing as well. It's not a simple RAW editor like Adobe Photoshop, because it has its own choices and needs.
Over the last 10 years, DaVinci Resolve has been widely used in film and television projects in Hollywood, as well as the broadcast industry. Mostly, this tool is used by colorists in color grading, retiming, correction, creative editing, etc. to achieve the requirements of the feature film project.
Another characteristic that distinguishes davinci resolve free video editor from other video editing and color grading software is its tight integration with the Blackmagic Design DaVinci family of cameras.
One key characteristic that sets DaVinci Resolve apart from other video editing solutions is its tight integration with the Blackmagic Design DaVinci family of cameras.
DaVinci Resolve has been the key color grading and editing application of choice for professionals working in the film and television industry worldwide since 2009. davinci resolve free video editor has been the key color grading and editing application of choice for professionals working in the film and television industry worldwide since 2009.
Resolve is not only used by colorists, it's also widely used by editors to perform a variety of editing functions, such as scene selection, effects, color correction, sound mixing, audio mastering and more.
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Davinci Resolve is a color grading application with an extensive array of features. It offers an easy workflow, powerful tools, and intuitive color presets. It supports a wide range of file formats, including Apple ProRes 4K and ProRes LT. DaVinci Resolve also offers a variety of creative grading and finishing tools, including multi-region color correcting and White Balance, Chromatic Aberration control, lens flare tools, waveform monitors, vibrant skin tools, and more. The app is available in both Windows and Mac OSX versions.
According to the Blackmagic website, the Blackmagic Resolve 18 was released on June 12, 2018. However, on their forum, it seems like early access purchasers can download a trial version to test the software. Users can upgrade once they pay a nominal fee of $299.
All Blackmagic Media products are incredibly expensive. Even the Blackmagic Resolve 18 is quite a hefty price for a grade application. If you're interested in editing, and color correcting sequences, I would highly recommend checking out the alternatives.
At it's most basic, davinci resolve free video editor is a post production color corrector. Larger, non-linear editing systems like Adobe Premiere and Avid Media Composer have color tools built in. The major difference is Premiere's color tool isn't the most advanced. Avid has full spectrum RGB (true color) specters (and they're easily adjustable) while Premiere has sRGB, which is what Windows displays. Premiere can also handle 10 bit data (24 bit true color), but it doesn't have any way to display that in the timeline. Resolve has full spectrum RGB, which can be adjusted to any color if you set it up to work that way.
Shoot your footage in full spectrum RGB (not sRGB) and the DaVinci Resolve color tools will work with it. If you shoot in sRGB, davinci resolve free video editor will work with sRGB (the default in Windows), but will not be able to process the full spectrum RGB data that you've gotten into your footage. Most of the time, you'll be dealing with sRGB and your color tools won't be able to work with it.
My advice is to buy the Premiere or Avid system with the color corrector built in. They will be able to correctly process your full spectrum RGB footage in either sRGB or RGB mode, and will work with your DaVinci Resolve color tools.
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