text:
Home Glock 23 Glock 23 vs Smith & Wesson 457 Comparison
Smith & Wesson 457
Glock 23 vs Smith & Wesson 457 Comparison
The 23 by Glock is a recoil operated semi-automatic Pistol produced during 1990 which sells new for $599.00 and has an estimated used valuation of $425.00. It's caliber is listed at .40 S&W with a muzzle velocity (MV) From 985 to 1325 ft/sec, muzzle energy (ME) From 355 to 500 ft-lb, and muzzle weight (WT) From 135 to 180 grains. The Glock 23 has fixed, white dot front, white square U rear sights, a integrated polymer grip, polymer frame, and a Glock single-action (SA) trigger. This handgun has a 13-round magazine with a maximum capacity of 14 rounds. The 23 has a recoil factor of 9.05 ft-lb and IDPA Power Factor of 174764.
Smith & Wesson 457 Summary
The 457 by Smith & Wesson is a recoil operated semi-automatic Pistol produced during 1996 - 2006 which sells new for $681.00 and has an estimated used valuation of $365.00. It's caliber is listed at .45 Automatic with a muzzle velocity (MV) From 830 to 1140 ft/sec, muzzle energy (ME) From 333 to 573 ft-lb, and muzzle weight (WT) From 165 to 230 grains. The Smith & Wesson 457 has white dot front, 2 dot rear sights, a straight backstrap wrap-around grip, aluminum alloy frame, and a double-action (DA/SA) trigger. This handgun has a 7-round magazine with a maximum capacity of 8 rounds. The 457 has a recoil factor of 8.45 ft-lb and IDPA Power Factor of 190470.
Title: Glock 23 Smith & Wesson 457
Caliber: .40 S&W .45 Automatic
Common Name: .40 Smith & Wesson Auto .45 Automatic Colt Pistol (ACP)
Production Years: 1990 1996 - 2006
Brand Name: Glock Smith & Wesson
Model Number: 23 457
Retail Price: $599.00 $681.00
Trigger: Glock single-action (SA) double-action (DA/SA)
Safety Type: 'Safe Action' trigger safety slide mounted safety/decock lever
Frame Material: polymer aluminum alloy
Grip Type: integrated polymer straight backstrap wrap-around
Sights: fixed, white dot front, white square U rear white dot front, 2 dot rear
Width: 1.18-in No Data
Length: 7.36-in 7.25-in
Height: 4.99-in No Data
Barrel: 4.01-in 3.75-in
Weight: 23.65-oz 29-oz
Beretta 9000S Type D vs Smith & Wesson 457 Comparison
Beretta 87 Cheetah vs Smith & Wesson 457 Comparison
Armscor-RIA TCM Rock Target vs Smith & Wesson 457 Comparison


text:
Rush 24/7 Morning Update: Ten-Year Saga
RUSH: A ten-year saga has thankfully come to a close. That's how long the Little Sisters of the Poor nuns have been fighting Obamacare's mandate to provide contraception in their health care plans. That goes against their Catholic faith. So finally, on Wednesday, the Supreme Court upheld President Trump's executive order exempting the nuns from Obama's mandate.
For 150 years, Little Sisters of the Poor nuns have run homes serving elderly people in poverty. The Democrats who passed Obamacare refused to acknowledge that as a religious order, the Little Sisters could not be forced to provide services that violate their religious beliefs.
Writing the majority opinion, Justice Clarence Thomas said the Trump administration had the authority to provide exceptions for religious and conscientious objections. He said the statute in plain language allowed the administration to make religious and moral exemptions.
Now, this decision wasn't even close, 7 to 2. Only the leftist Justices Ginsburg and Sotomayor dissented. They're mad at the Court's "zeal to secure religious rights to the Nth degree."
But they're really mad at the First Amendment, which states that Congress cannot make laws that infringe on religious freedoms.
The right thing was done, even though it took ten years. It's clear to anyone who respects our Constitution that Obama never had the power to impose this mandate in the first place.
National Review: Supreme Court Rules in Favor of Little Sisters in Contraception-Mandate Case

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: find an element by style with jquery How could I find an element in an HTML document using it's style properties?
here is an example:
HTML:

e1

div

e2

e3

e4

e5

The question is how do i select, for example..: the element
 with z-index=4.
and how to select all div's with z-index=8 ...

A: If the style is set inline, you can use an attribute selector:
 $('li[style*="z-index:4"]') //returns any li's with z-index = 4

see here. The advantage to this method is that it is very fast.
If the style is set via a stylesheet, you can access it this way:
 var elem;
 var elems = $("li");

 for(var i=0; i<elems.length; i++) {
    if($(elems[i]).css('z-index') == '4') {
        elem = elems[i];  //assign elem we found
        break;            //exit loop early
    }
 }

Note Webkit browsers, (Safari, Chrome, etc), will not return a z-index value unless it is positioned as well. See this example
Also, for loop is still faster than .filter()

A: There isn't a style selector (how would it work, the style is a combination of inherited and explicitly stated rules), but you can filter (demo):
var things = $("li").filter(function() {
  return $(this).css('z-index') == '4';
});

Now if you are using z-index to attach a piece of data to an HTML element you might have more luck using data attributes (cleaner, and searchable by Sizzle).  One interpretation might look like this: (demo)

div

e1

e2

e3

e4

e5

redpajama_set_name: RedPajamaStackExchange


I'm involved now in a project to restore a 1800's AME Mt. Pisgah cemetery in Elsinboro Twp NJ. It is a embarrasment what kind of shape I found it. The African Americans that are buried here were involved with or have family involved with the Underground Railroad or Veterans of all the wars including the Civil War.


text:
Q: C++, Reading and Saving large amount of numbers I have to solve a problem and i need your help, so thanks in advance. As you can see, title says almost everything. There is a file "domes.in" (which is a textfile), which contains 10.000.000 pairs of integers From (1 - 100.000), so in the first part, I have to save all those numbers, in less than 1 sec.
First, I have tried creating an int array of 20.000.000 places, so that i could save the numbers one by one The size of the array is so big that the program does not respond.
Before you read the code, read the task, but please just answer my question and do not post any code, because it is a problem of an online competition and i would like to write it by myself.
#include <fstream>
using namespace std;

int main()
{
    int n, m, a, b;        //N(Value)    M(Pairs)
    int t = 0;
    int r = 0;
    int j;

    ifstream infile("domes.in");
    infile >> n >> m;     //N = 100.000    M = 10.000.000
    int domes[m*2];

    for (j=0; j<m; j++)      //For 1 to 10.000.000
    {
        infile >> a >> b;    //Save 10.000.000 Integers
        domes[t++] = a;
        domes[t++] = b;
    }
    for (j=1; j<=n; j++)          //For J = 1 - 100.000
    {
        int i=0;
        for (int k=0; k<t; k++)   //If the point J is appeared
            if (domes[k] == j)    //+1 Link
                i++;

        if (i == 1)               //If Links < 2  (of number J)
            r++;                  //+1 Point is connected with less than 2 points
    }                             //Else move on to ++J
    infile.close();

    ofstream outfile("domes.out");
    outfile << r;
    return 0;
}

Now it seems that it might work, but again the program doesn't respond (In runtime, in raises an error, which says that the program stopped running. There are any other build errors or mistakes in the code).
What am i doing wrong?
The TextFile:
100000 10000000
2344 3444
3345 4564
5566 9455     //Random integers up to 100.000
//..........
//Edit: I  Removed the struct "DOMES" from the code, since it was the same thing
The Task:    I will explain it with my own words, because it is complicated.  The are   100.000 points on the map. These points, are connected (There are 10.000.000 links). I have to find the points which have only a single connection (Each Point has at least 1 connection).
For example:
5 4    (N = 5,  M = 4)    N(Max Value)   M(Pairs)
1 2
2 3
4 5
3 4
There are 2 points that have a single connection:   1 and 5  (1 is connected with 2) and (5 is connected with 4). The rest of the points are connected at least twice:
2:  1-2, 2-3
3:  2-3, 3-4
4:  4-3, 4-5

A: The first question is where the cost of your algorithm lies, which is most probably not related to the reading of the input. If I understand your algorithm, you are reading all of the numbers (node id) into memory, then you iterate over the links, you pick one node and attempt to find if there are more links that include that node. That algorithm is O(N^2), which for a large number becomes a huge number of operations.
If what you need to know is how many of the nodes have a single connection, all you need to maintain in memory is one entry per node (100k). How much information do you need to keep per node? Little, for example to only detect which nodes have a single connection you can store the number of times the node has been seen (0 -> never seen). Read one line at a time, increment the counts for both nodes (cost: O(N) with N being the number of links). At the end of the process, scan the entries and print out those for which the count is 1 (cost: O(M), where M is the number of nodes). The total cost of the algorithm is linear in the sum of the number of nodes + number of edges, with linear space requirements on the number o nodes.
Compare that with the original approach: space 400k vs. 160M; operations 10M vs. 10^14.
In most programming contests the goal is not to shave the cost of the operations, but to improve the asymptotic complexity of the solution. Naive solutions that seem to work for small inputs won't work for large ones. O(N^2) for a problem of size 10 is basically 100 operations... almost a constant factor! But as the problem increases, with size 100 it becomes 10k, with size 1000 it is 1M...

A: When it comes to I/O of large data, C++ streams are slower than their C relatives. I am sure you can speed the writing up with fopen() and fwrite() from <cstdio> a lot. By doing this you could write the whole array in one go and probably wouldn't have the crash issue.

A: Not sure whether this is will be useful for an online competition. However, for fast access to files I would try memory mapped files. (Use mmap() if your platform supports POSIX, or CreateFileMapping and MapViewOfFile on Windows) Basically you then write to file as you would write to memory.


text:
\section{Introduction}

Let ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$ be the ring of integers modulo 2 and 4
respectively. Let ${\mathbb{Z}}_2^n$ denote the set of all binary vectors of
length $n$ and let ${\mathbb{Z}}_4^n$ be the set of all quaternary vectors of
length $n$. Any non-empty subset $C$ of ${\mathbb{Z}}_2^n$ is a binary code
and a subgroup of ${\mathbb{Z}}_2^n$ is called a {\it binary linear code} or a
{\it ${\mathbb{Z}}_2$-linear code}. Equivalently, any non-empty subset ${\cal
C}$ of ${\mathbb{Z}}_4^n$ is a quaternary code and a subgroup of ${\mathbb{Z}}_4^n$ is
called a {\it quaternary linear code}.

Quaternary linear codes can be viewed as binary codes under the Gray map defined as $\phi(0)=(0,0),\ \phi(1)=(0,1), \ \phi(2)=(1,1),\ \phi(3)=(1,0)$. If ${\cal C}$ is a quaternary linear code, then the binary code $C=\phi({\cal C})$ (coordinatewise extended) is said to be a {\em ${\mathbb{Z}}_4$-linear} code.
The notions of dual code of a quaternary linear code ${\cal C}$, denoted by ${\cal C}^{\perp}$, self-orthogonal code (when ${\cal C} \subseteq {\cal C}^{\perp}$) and self-dual code (when ${\cal C}={\cal C}^{\perp}$)
are defined in the standard way (see \cite{MacW}) in terms of the usual inner product for quaternary vectors (see \cite{Sole}). Since in general the binary code $C=\phi({\cal C})$ is not linear, it need not have a dual. However, the binary code $C_{\perp}=\phi({\cal C}^\perp)$ is called the {\it ${\mathbb{Z}}_4$-dual} of $C=\phi({\cal C})$.

Since 1994, quaternary linear codes have became significant due to
its relationship to some classical well-known binary codes as the
Nordstrom-Robinson, Kerdock, Preparata, Goethals or Reed-Muller
codes (see \cite{Sole}). It was proved that the Kerdock code and the
Preparata-like code are ${\mathbb{Z}}_4$-linear codes and, moreover, the
${\mathbb{Z}}_4$-dual code of the Kerdock code is the Preparata-like code.
Lately, also some families of quaternary linear codes, called $QRM$
and $ZRM$, related to the Reed-Muller codes have been studied in
\cite{QRM} and \cite{ZRM}, respectively.

\medskip
Additive codes were first defined by Delsarte in $1973$ in terms of
association schemes (see \cite{del}, \cite{lev}). In general, an
additive code, in a translation association scheme, is defined as a
subgroup of the underlying abelian group. In the special case of a
binary Hamming scheme, that is when the underlying abelian group is
of order $2^n$, the only structures for the abelian group are those
of the form ${\mathbb{Z}}_2^\alpha\times {\mathbb{Z}}_4^\beta$, with $\alpha +
2\beta=n$. Therefore, the subgroups ${\cal C}$ of
${\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ are the only additive codes in a
binary Hamming scheme. In order to distinguish them from additive
codes over finite fields (see \cite{BG}, \cite{Bier}, \cite{BBrou},
\cite{LP}), from now on we will call them {\em $\Z_2\Z_4$-additive
codes}.

The binary image of a $\Z_2\Z_4$-additive code under the extended Gray
map defined in Section \ref{sec:Definitions} is called {\it
$\Z_2\Z_4$-linear code}. In \cite{Adds} and \cite{PropIT}, binary perfect 1-error
correcting codes (or 1-perfect codes) which are $\Z_2\Z_4$-linear codes
are described and all such 1-perfect codes are characterized. More
examples, such as extended 1-perfect  and Hadamard codes which are
$\Z_2\Z_4$-linear codes are studied in subsequent papers (see
\cite{ExtPerf}, \cite{Kro01}, \cite{Hadam}, \cite{PheRif}).  Some notorious codes, e.g.
Kerdock-like and Preparata-like codes, can have a ${\mathbb{Z}}_4$-linear
structure (see \cite{Sole}), but they cannot have a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear
structure with non-empty binary part (see \cite{PrepKerd}).

\medskip
As we have seen, the $\Z_2\Z_4$-additive codes belong to the more
general family of additive codes. However, notice that one could
think of other families of codes with an algebraic structure that
also include the $\Z_2\Z_4$-additive codes; such as mixed group codes
and translation invariant propelinear codes.

{\em Mixed group codes} are defined as subgroups of a group of type
$G=G_1\times\cdots\times G_r$, where $G_i$ is a finite abelian group
for all $i=1,\ldots,r$ (see \cite{Hede}, \cite{Lind}).
Since any finite abelian group has a
factorization in cyclic groups, we can also think about a mixed
group code as a subgroup of ${\mathbb{Z}}_{i_1}\times\cdots\times{\mathbb{Z}}_{i_s}$,
where the indices $i_1,\ldots,i_s$ are not necessarily different. If
we are interested in a binary version of these codes, we need a
one-to-one mapping $\phi$ from ${\mathbb{Z}}_{i_j}$ to ${\mathbb{Z}}_2^m$ (where
$2^m\geq i_j$) for all $j=1,\ldots,s$. In \cite{Codis2k}, it was
shown that the indices $i_1,\ldots,i_s$ must be all even, if we want
to use a Gray map $\phi$ which has the classical property that
$d(\phi(i),\phi(i+1))=1$, where $d(\cdot,\cdot)$ is the Hamming
distance between binary vectors. This Gray map is unique, up to
coordinate permutation, if the binary image is also Hamming
compatible. Moreover, it was also proved that if the binary image of
a subgroup of ${\mathbb{Z}}_{i_1}\times\cdots\times{\mathbb{Z}}_{i_s}$, using such Gray
map, is a 1-perfect code, then $i_j\in\{2,4\}$, for all
$j=1,\ldots,s$ (i.e. it is also a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear code).

{\em Translation invariant propelinear codes} were first defined in 1997
(see \cite{PropIT}, \cite{Propes}). In \cite{PropIT}, it was also proved that all such
binary codes are group-isomorphic to subgroups of
${\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta\times Q_8^k$, where $Q_8$ is the
non-abelian quaternion group on eight elements. Hence, abelian
translation invariant propelinear codes are exactly all the
${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear codes.

\medskip
Most of the concepts on $\Z_2\Z_4$-additive codes described in this
paper have been implemented by the authors as a new package in {\sc
Magma} (see \cite{Magma}). {\sc Magma} is a software package
designed to solve computationally hard problems in algebra, number
theory, geometry and combinatorics. Currently it supports the basic
facilities for linear codes over integer residue rings and Galois
rings; moreover, it also supports functions for additive codes over
a finite field, which are a generalization of the linear codes over
a finite field (see \cite[Chapter 119, 120]{M1}). However, it does
not include functions to work with $\Z_2\Z_4$-additive codes. For this
reason, a beta version of this new package for $\Z_2\Z_4$-additive codes
and the manual with the description of all functions can be
downloaded from the web page {\tt http://www.ccg.uab.cat}. For any
comment or further information about this package, you can send an
e-mail to {\em support-ccg@deic.uab.cat}.

\medskip
The aim of this paper is a general study of $\Z_2\Z_4$-additive codes
and the corresponding $\Z_2\Z_4$-linear codes. It is organized as
follows. In Section \ref{sec:Definitions}, we give the definition of
$\Z_2\Z_4$-additive and $\Z_2\Z_4$-linear codes, we find which are the
fundamental parameters, and we also discuss about the automorphism
groups of these codes. In Section \ref{sec:matrices}, we deduce a
standard form for generator matrices of $\Z_2\Z_4$-additive codes.
Section \ref{sec:duality} is devoted to study duality for
$\Z_2\Z_4$-additive codes defining the appropriate inner product. In
Section \ref{sec:parityCheck}, we show how the generator and parity
check matrices are related and we also compute the parameters of the
dual code. Finally, in Section \ref{sec:selfdual}, we give some
conditions for self-duality.

\section{Definitions}
\label{sec:Definitions}

From~now on, we will focus on $\Z_2\Z_4$-additive codes ${\cal C}$, which
are subgroups of ${\mathbb{Z}}_2^{\alpha}\times{\mathbb{Z}}_4^{\beta}$. We will take an
extension of the usual Gray map: $\Phi:
{\mathbb{Z}}_2^{\alpha}\times{\mathbb{Z}}_4^{\beta} \longrightarrow {\mathbb{Z}}_2^{n}$, where
$n=\alpha+2\beta$, given by
$$\begin{array}{lc}
\Phi(x,y) = (x,\phi(y_1),\ldots,\phi(y_\beta))\\
 \hspace{1truecm}\forall x\in{\mathbb{Z}}_2^\alpha,\;\forall y=(y_1,\ldots,y_\beta)\in {\mathbb{Z}}_4^\beta;
\end{array}$$
where $\phi:{\mathbb{Z}}_4 \longrightarrow {\mathbb{Z}}_2^{2}$ is the usual Gray map,
that is, $$\phi(0)=(0,0),\ \phi(1)=(0,1), \ \phi(2)=(1,1), \
\phi(3)=(1,0).$$ This Gray map is an isometry which transforms Lee
distances defined in the $\Z_2\Z_4$-additive codes ${\cal C}$ over
${\mathbb{Z}}_2^{\alpha}\times{\mathbb{Z}}_4^{\beta}$ to Hamming distances defined in
the binary codes $C=\Phi({\cal C})$. Note that the length of the binary code
$C$ is $n=\alpha+2\beta$.

\medskip
Since ${\cal C}$ is a subgroup of  ${\mathbb{Z}}_2^{\alpha}\times
{\mathbb{Z}}_4^{\beta}$, it is also isomorphic to an abelian structure like
${\mathbb{Z}}_2^{\gamma}\times {\mathbb{Z}}_4^{\delta}$. Therefore, ${\cal C}$ is of type $2^\gamma 4^\delta$ as a group, it has
$|{\cal C}|=2^{\gamma+2\delta}$ codewords and the number of order two codewords
in ${\cal C}$ is $2^{\gamma+\delta}$.

Let $X$ (respectively $Y$) be the set of ${\mathbb{Z}}_2$ (respectively
${\mathbb{Z}}_4$) coordinate positions, so $|X|=\alpha$ and $|Y|=\beta$. Unless otherwise stated, the set $X$ corresponds to the first $\alpha$ coordinates and $Y$ corresponds to the last $\beta$ coordinates. Call
${\cal C}_X$ (respectively ${\cal C}_Y$) the punctured code of ${\cal C}$ by deleting the coordinates outside $X$
(respectively $Y$). Let ${\cal C}_b$ be the subcode of
${\cal C}$ which contains all order two codewords and let $\kappa$ be
the dimension of $({\cal C}_b)_X$, which is a binary linear code. For the case
$\alpha=0$, we will write $\kappa=0$.

Considering all these parameters, we will say that ${\cal C}$
(or equivalently $C=\Phi({\cal C})$) is of type
$(\alpha,\beta;\gamma,\delta;\kappa)$. Notice that ${\cal C}_Y$ is a
quaternary linear code of type $(0,\beta;\gamma_Y,\delta;0)$, where
$0\leq \gamma_Y \leq \gamma$, and ${\cal C}_X$ is a binary linear code of
type $(\alpha,0;\gamma_X,0;\gamma_X)$, where $\kappa \leq \gamma_X \leq \gamma$.

\begin{defi} Let ${\cal C}$ be a $\Z_2\Z_4$-additive code, that is a
subgroup of ${\mathbb{Z}}_2^{\alpha}\times {\mathbb{Z}}_4^{\beta}$. We say that the
binary image $C=\Phi({\cal C})$ is a {\em $\Z_2\Z_4$-linear code} of
length $n=\alpha+2\beta$ and type
$(\alpha,\beta;\gamma,\delta;\kappa)$, where $\gamma$, $\delta$ and $\kappa$  are defined as above.
\end{defi}

Note that ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear codes are a generalization of
binary linear codes and ${\mathbb{Z}}_4$-linear codes. When $\beta=0$, the
binary code $C={\cal C}$ corresponds to a binary linear code. On the
other hand, when $\alpha=0$, the $\Z_2\Z_4$-additive code ${\cal C}$ is a
quaternary linear code and its corresponding binary code
$C=\Phi({\cal C})$ is a ${\mathbb{Z}}_4$-linear code.

\medskip Two $\Z_2\Z_4$-additive codes ${\cal C}_1$ and ${\cal C}_2$ both of type
$(\alpha,\beta;\gamma,\delta;\kappa)$ are said to be {\it
monomially equivalent}, if one can be obtained from the other by permutating
the coordinates and (if necessary) changing the signs of certain
${\mathbb{Z}}_4$ coordinates. Two $\Z_2\Z_4$-additive codes are said to be {\it
permutation equivalent} if they differ only by a permutation of
coordinates. The {\it monomial automorphism group} of a
$\Z_2\Z_4$-additive code ${\cal C}$, denoted by  $MAut({\cal C})$,
is the group generated by all permutations and sign-changes
of the ${\mathbb{Z}}_4$ coordinates that preserve the set of codewords of ${\cal C}$, while
the {\it permutation automorphism group} of ${\cal C}$, denoted by $PAut({\cal C})$,
is the group generated by all permutations that preserve the set of codewords of ${\cal C}$ (see \cite{Pless}).

If two $\Z_2\Z_4$-additive codes ${\cal C}_1$ and ${\cal C}_2$ are monomially
equivalent, then, after the Gray map, the corresponding
$\Z_2\Z_4$-linear codes $C_1=\Phi({\cal C}_1)$ and $C_2=\Phi({\cal C}_2)$ are
isomorphic as binary codes. Note that the inverse statement is not
always true.

\section{Generator matrices of $\Z_2\Z_4$-additive codes}
\label{sec:matrices}

Let ${\cal C}$ be a $\Z_2\Z_4$-additive code. Although ${\cal C}$ is not a
free module, every codeword is uniquely expressible in the form  $$\displaystyle
\sum_{i=1}^{\gamma}\lambda_iu_i+\sum_{j=\gamma+1}^{\gamma+\delta}\mu_jv_j,$$
where $\lambda_i \in {\mathbb{Z}}_2$ for $1\leq i\leq \gamma$,
$\mu_j\in{\mathbb{Z}}_4$ for $\gamma+1\leq j\leq \gamma+\delta$ and $u_i,
v_j$ are vectors in ${\mathbb{Z}}_2^\alpha\times
{\mathbb{Z}}_4^\beta$ of order two and order four, respectively. The vectors $u_i,v_j$ give us a
generator matrix ${\cal G}$ of size $(\gamma+\delta)\times (\alpha+\beta)$ for
the code ${\cal C}$. Moreover, we can write ${\cal G}$ as \begin{equation} \label{eq:matrixG}
    {\cal G}= \left (\begin{array}{c|c} B_1&2B_3\\ \hline B_2&Q\end{array}\right ),
\end{equation} %
where $B_1,B_2,B_3$ are matrices over ${\mathbb{Z}}_2$ of size $\gamma\times
\alpha$,  $\delta\times \alpha$ and $\gamma\times \beta$,
respectively; and $Q$ is a matrix over ${\mathbb{Z}}_4$ of size
$\delta\times \beta$ with quaternary row vectors of order four.

In \cite{Sole}, it was shown that any quaternary linear
code of type $(0,\beta;\gamma,\delta;0)$ is permutation equivalent
to a quaternary linear code with a generator matrix of the form
\begin{equation} \label{eq:QaryStandardForm} {\cal G}_S=\left ( \begin{array}{|ccc}
2T & 2I_{\gamma} & {\mathbf{0}}\\ \hline
S & R & I_\delta \end{array} \right ), \end{equation} where $R,T$ are
matrices over ${\mathbb{Z}}_2$ of size $\delta\times\gamma$ and
$\gamma\times(\beta-\gamma-\delta)$, respectively; and $S$ is a
matrix over ${\mathbb{Z}}_4$ of size $\delta\times(\beta-\gamma-\delta)$.
In this section, we will generalize this result for $\Z_2\Z_4$-additive codes, so
we will give a canonical generator matrix for these codes (see \cite{BF06}).

First, notice
that changing ones by twos in the coordinates over ${\mathbb{Z}}_2$, we can
see the $\Z_2\Z_4$-additive codes as quaternary linear codes. Let
$\chi$ be the map from ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_4$, which is the usual
inclusion from the additive structure in ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_4$:
$\chi(0)=0, \ \chi(1)=2.$ This map can be extended to the map
$(\chi, Id): {\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta \longrightarrow
{\mathbb{Z}}_4^\alpha\times{\mathbb{Z}}_4^\beta$, which will also be denoted by
$\chi$.

\medskip
\begin{theo}\label{prop:StandardForm}
  Let ${\cal C}$ be a $\Z_2\Z_4$-additive code of type
  $(\alpha,\beta;\gamma,\delta;\kappa)$. Then, ${\cal C}$ is
  permutation equivalent to a $\Z_2\Z_4$-additive code with canonical generator matrix
  of the form
 \begin{equation} \label{eq:StandardForm}
{\cal G}_S= \left ( \begin{array}{cc|ccc}
I_{\kappa} & T_b & 2T_2 & {\mathbf{0}} & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}} & 2T_1 & 2I_{\gamma-\kappa} & {\mathbf{0}}\\
\hline {\mathbf{0}} & S_b & S_q & R & I_{\delta} \end{array} \right ),
\end{equation} \noindent where $T_b, T_1, T_2, R, S_b$ are matrices over ${\mathbb{Z}}_2$
and $S_q$ is a matrix over ${\mathbb{Z}}_4$.
\end{theo}

\begin{demo} Since $\kappa$ is the dimension of the matrix $B_1$ over ${\mathbb{Z}}_2$
given in (\ref{eq:matrixG}), the code ${\cal C}$ has a generator matrix of the form
$$ \left (
\begin{array}{cc|c}
I_{\kappa} & \bar{B_1} & 2\bar{B_3}\\
{\mathbf{0}} & {\mathbf{0}} & 2\bar{B_4}\\ \hline
\vspace{-.35truecm} & &  \\
{\mathbf{0}} & \bar{B_2} & \bar{Q}
\end{array} \right),
$$ %
where $\bar{B_1},\bar{B_2},\bar{B_3}$ and $\bar{B_4}$ are matrices
over ${\mathbb{Z}}_2$ of size $\kappa\times(\alpha-\kappa)$, $\delta\times(\alpha-\kappa)$,
$\kappa\times\beta$ and $(\gamma-\kappa)\times\beta$, respectively;
and $\bar{Q}$ is a matrix over ${\mathbb{Z}}_4$
of size $\delta\times\beta$.

The quaternary linear code ${\cal C}^{-}$ of type
$(0,\alpha-\kappa+\beta;\gamma-\kappa,\delta;0)$ generated by
the matrix $$ \left (\begin{array}{|cc}
{\mathbf{0}} & 2\bar{B_4}\\ \hline
\vspace{-.35truecm} &  \\
2\bar{B_2} & \bar{Q}
\end{array} \right) $$
is permutation equivalent to a quaternary linear code with
generator matrix of the form $${\cal G}^{-}=\left(
\begin{array}{|cccc}
{\mathbf{0}} & 2T_1 & 2I_{\gamma-\kappa} & {\mathbf{0}}\\ \hline
2S_b &  S_q & R & I_\delta \end{array} \right ),$$ where the
permutation of coordinates fixes the first $\alpha-\kappa$
coordinates (see \cite{Sole} or (\ref{eq:QaryStandardForm})).
So, the quaternary linear code $\chi({\cal C})$ generated by the matrix
$$ \left (
\begin{array}{|ccc}
2I_{\kappa} & 2\bar{B_1} & 2\bar{B_3}\\
{\mathbf{0}} & {\mathbf{0}} & 2\bar{B_4}\\ \hline
\vspace{-.35truecm} & &  \\
{\mathbf{0}} & 2\bar{B_2} & \bar{Q} \end{array} \right)
$$ %
is permutation equivalent to a quaternary linear code with
generator matrix of the form $${\cal G}_\chi= \left (
\begin{array}{|ccccc}
2I_{\kappa} & 2T_b & 2T_2 & {\mathbf{0}} & {\mathbf{0}}\\
{\mathbf{0}} & {\mathbf{0}} & 2T_1 & 2I_{\gamma-\kappa} & {\mathbf{0}}\\ \hline
{\mathbf{0}} & 2S_b & S_q & R & I_{\delta} \end{array} \right ).$$

Finally, ${\cal C}$ is permutation equivalent to a
$\Z_2\Z_4$-additive code with generator matrix $\chi^{-1}({\cal G}_\chi)={\cal G}_S$.
\end{demo}

\begin{ex} \label{example:C1} Let ${\cal C}_1$ denote the
$\Z_2\Z_4$-additive code of type $(1,3;1,2;1)$ with generator matrix
  $${\cal G}= \left (\begin{array}{c|c c c} 1 & 2 & 2 & 2\\ \hline
      0 & 1 & 1 & 0\\
      1 & 1 & 2 & 3
\end{array}\right ).$$ The code ${\cal C}_1$ can also be generated by the matrix
$$\left (\begin{array}{c|c c c} 1 & 2 & 2 & 2\\ \hline
      0 & 1 & 1 & 0\\
      0 & 1 & 0 & 3
\end{array}\right ).$$
The quaternary linear code ${\cal C}^{-}$ generated by $\left (\begin{array}{|c c c} \hline
                                              1 & 1 & 0\\
                                              1 & 0 & 3
\end{array}\right )$ is permutation equivalent to a quaternary linear code with
generator matrix ${\cal G}^{-} =\left (\begin{array}{|c c c} \hline
                                               1 & 1 & 0\\
                                               3 & 0 & 1
\end{array}\right ).$
So, the quaternary linear code $\chi({\cal C})$ generated by
$$\left (\begin{array}{|c c c c} 2 & 2 & 2 & 2\\ \hline
                                             0 & 1 & 1 & 0\\
                                             0 & 1 & 0 & 3
\end{array}\right )$$ is permutation equivalent to a quaternary
linear code with generator matrix
$${\cal G}_{\chi}=\left (\begin{array}{|c c c c} 2 & 2 & 0 & 0\\ \hline
                                 0 & 1 & 1 & 0\\
                                 0 & 3 & 0 & 1
\end{array}\right ).$$ Therefore, the code ${\cal C}_1$ is
permutation equivalent to a $\Z_2\Z_4$-additive code with canonical generator
matrix $${\cal G}_S=\chi^{-1}({\cal G}_{\chi})=\left (\begin{array}{c|ccc} 1 & 2 & 0 & 0\\
\hline
                                     0 & 1 & 1 & 0\\
                                     0 & 3 & 0 & 1
\end{array}\right ).$$
\end{ex}

\begin{ex} \label{example:C5} Let $\mathcal{C}_2$ be a
$\Z_2\Z_4$-additive code of type $(3,4;3,1;3)$ with generator matrix
$$\left (\begin{array}{ccc|cccc}1 & 0 & 0  &  2 & 2 & 0 & 0 \\
                                1 & 1 & 1  &  2 & 2 & 2 & 2 \\
                                1 & 1 & 0  &  2 & 2 & 0 & 0 \\\hline
                                1 & 1 & 1  &  1 & 1 & 1 & 1 \\
\end{array}\right ).$$
By Theorem \ref{prop:StandardForm}, ${\cal C}_2$ is
  permutation equivalent to a $\Z_2\Z_4$-additive code with canonical generator matrix
$$\left (\begin{array}{ccc|cccc}1 & 0 & 0  &  2 & 2 & 0 & 0 \\
                                0 & 1 & 0  &  0 & 0 & 0 & 0 \\
                                0 & 0 & 1  &  2 & 2 & 0 & 0 \\\hline
                                0 & 0 & 0  &  1 & 1 & 1 & 1 \\
\end{array}\right ).$$
\end{ex}

\section{Duality of $\Z_2\Z_4$-additive codes}
\label{sec:duality}

For linear codes over finite fields or finite rings there exists
the well-known concept of duality. In this section, we will study this
concept for $\Z_2\Z_4$-additive codes. First, we will show that the inner product of elements
of a finite abelian group can be uniquely defined. Then, considering the finite group
${\mathbb{Z}}_2^\alpha \times {\mathbb{Z}}_4^\beta$ we will define the notions of duality, as the additive dual code and
the $\Z_2\Z_4$-dual code, for $\Z_2\Z_4$-additive codes and its corresponding $\Z_2\Z_4$-linear codes, respectively.

\bigskip
Given a finite abelian group $(G,+)$ of exponent
$m\geq1$ (so, for each $b\in G$ we have $m b=0$), we call {\em dual
group} of $G$, denoted by $\widehat{G}$, the group of ho\-mo\-mor\-phisms
from $G$ into ${\mathbb{Z}}_m$, $\widehat{G}=Hom(G,{\mathbb{Z}}_m)$.

\begin{ex}\label{zm} Let $(G,+)$ be the cyclic group of order four
with generator $a$, so $G=\{a,2a,3a,4a=0\}$.
If we know $\varphi(a)$, where $\varphi\in\widehat{G}$, we will know the image
of any element in $G$, because for any $b\in G$ we have $b=i a$
and $\varphi(b)=\varphi(i a)=i \varphi(a)\in {\mathbb{Z}}_4$. So, there are four
different homomorphisms that we can define over $G$: $$
\begin{array}{r|rrrr}
&\varphi(0)&\varphi(a)&\varphi(2a)&\varphi(3a)\\
\hline
\varphi_0&0&0&0&0\\
\varphi_1&0&1&2&3\\
\varphi_2&0&2&0&2\\
\varphi_3&0&3&2&1 \end{array} $$

Note that, in general, for any cyclic group $(G,+)$ of order $m$
with generator $a$, we can construct all the possible
homomorphisms as $\varphi_k(i a)=k i\in {\mathbb{Z}}_m$. Note also that
$\varphi_k(b) + \varphi_s (b)=\varphi_{k + s}(b)$, for any $b\in G$.
\end{ex}

It is well-known that ($\widehat{G},\cdot)$ is an abelian group by
using the operation $(\varphi \cdot \lambda )(g)=\varphi(g)+\lambda(g)$, where
$\varphi,\lambda \in \widehat{G}$ and $g\in G$ (see \cite{lang}). The
group $\widehat{G}$ has the same cardinality as $G$ and both
groups are isomorphic (see \cite{lang}), but there is no a
canonical (or natural) isomorphism from $G$ to $\widehat{G}$.

\bigskip Assume $G$ is a cyclic group of order $m$ and fix a
generator $a\in G$. Any homomorphism $\varphi\in
\widehat{G}=Hom(G,{\mathbb{Z}}_m)$ is defined knowing $\varphi(a)$. If
$\varphi(a)=k\in {\mathbb{Z}}_m$, this homomorphism will be denoted by $\varphi_k$
and, for any element $b=ia \in G$, $\varphi_k(b)=\varphi_k(i a)=i \varphi_k(a)=i k\in {\mathbb{Z}}_m$.
So, we can define an isomorphism $G
\longrightarrow \widehat{G}$, such that for all $c\in G$ we
have $c=ja \mapsto \varphi_j$. Note that this isomorphism depends on
the fixed generator $a\in G$.

\bigskip Let $G_1, G_2$ be two abelian groups of exponent $m\geq
1$. A \textit{bilinear map} of $G_1\times G_2$ into ${\mathbb{Z}}_m$ is a
map $$ \begin{array}{ccc}
G_1 \times G_2 &\longrightarrow & {\mathbb{Z}}_m\\
(x_1,x_2)&\longmapsto & \langle x_1,x_2\rangle\end{array} $$ such
that for $x_1\in G_1$ the function $x_2 \mapsto \langle
x_1,x_2\rangle$ and for $x_2\in G_2$ the function $x_1 \mapsto
\langle x_1,x_2\rangle$ are homomorphisms.

Let $G$ be an abelian group of exponent $m\geq 1$. A special case of
bilinear map is \begin{equation} \label{bm} G\times \widehat{G} \longrightarrow
{\mathbb{Z}}_m,\end{equation} where $(b,\varphi) \mapsto \varphi(b) \in {\mathbb{Z}}_m$, for all
$b\in G$ and $\varphi \in \widehat{G}$. Another special case of
bilinear map is the so called \textit{inner product} in $G$ given by
\begin{equation} \label{bm2} G\times G \longrightarrow {\mathbb{Z}}_m,\end{equation} where
$(b,c)\mapsto \varphi(b) \in {\mathbb{Z}}_m$, $\widehat{\varphi}:
G\longrightarrow \widehat{G}$ is a fixed isomorphism and
$\varphi=\widehat{\varphi}(c)$, for all $b,c\in G$.

Note that although the bilinear map given by (\ref{bm})
is canonically defined, the inner product defined by
(\ref{bm2}) depends on the particular isomorphism $\widehat{\varphi}$
from $G$ to $\widehat{G}$ that we use.

\bigskip Assume again $G$ is a cyclic group of order $m$ and fix a
generator $a\in G$. The inner product in $G$ is defined
uniquely by \begin{equation}\label{inn}(b,c) \mapsto \langle b,c\rangle =\varphi_j(b)=\varphi_j(i a)=j i \in {\mathbb{Z}}_m,\end{equation}
where $b=i a \in G$, $c=j a \in G$ and $\varphi_j \in \widehat{G}=Hom(G,{\mathbb{Z}}_m)$.

Let $G'$ be a subgroup of $G$ generated by an element $a'\in G'$ of order $t$,
where $t \mid m$.
The dual group of $G'$ could be considered as $Hom(G',{\mathbb{Z}}_t)$ or
$Hom(G',{\mathbb{Z}}_m)$ depending on whether the exponent of
$G'$ is $t$ or $m$, respectively.
In both cases a generator $a'$ in $G'$ is send to an
element of order $t$ in ${\mathbb{Z}}_t$ or ${\mathbb{Z}}_m$, respectively. This situation can be represented by
$$\begin{array}{ccccc}G' &\longrightarrow &
{\mathbb{Z}}_t
&\longrightarrow &{\mathbb{Z}}_m\\
a' & \longmapsto &1 & \longmapsto&\bar{s} \end{array} $$ where
$\bar{s}\in {\mathbb{Z}}_m$ is an element of order $t$.
Then, after fixing a generator $a' \in G'$ and an element
$\bar{s}\in {\mathbb{Z}}_m$ of order $t$, the inner product of elements of
$G'$ seen as elements in $G$ is defined uniquely by
$$(b,c) \mapsto \langle b,c\rangle_m = \bar{s} \langle
b,c\rangle_t =\bar{s}
\varphi_j(b)=\bar{s} \varphi_j(i a')=\bar{s} j i \in {\mathbb{Z}}_m,$$ where $b=i
a'$, $c=j a'$ and $\varphi_j\in Hom(G',{\mathbb{Z}}_t)$.

\bigskip It is well-known (see~\cite{lang}) that if $G$ is a
finite abelian group, expressed as a product $G=G_1\times G_2$,
then $\widehat{G}$ is isomorphic to $\widehat{G_1}\times
\widehat{G_2}$ under the mapping $\widehat{G_1}\times
\widehat{G_2} \longrightarrow \widehat{G}$, where the element
$(\lambda_1,\lambda_2)\in \widehat{G_1}\times \widehat{G_2}$ is
transformed into an element in $\widehat{G}$ such that for all $(x_1,x_2)\in G$
$$(\lambda_1,\lambda_2)(x_1,x_2)=\lambda_1(x_1)+\lambda_2(x_2).$$

Moreover, any finite abelian group $G$ is isomorphic to a product of cyclic
groups, specifically, we can write $$ G\cong  {\mathbb{Z}}_{m_1} \times
{\mathbb{Z}}_{m_2}\times \cdots \times {\mathbb{Z}}_{m_k},$$ where $ 1 < m_1 \,|\,m_2
\,|\,\cdots \,|\,m_k$.
Therefore, we have that the inner product, defined by
(\ref{inn}) in a cyclic group,
can be extended to any finite abelian group in the
following way:

\begin{prop}\label{defi} Let $(G,+)$ be a finite abelian group of
exponent $m$ and consider the decomposition $$ G\cong  {\mathbb{Z}}_{m_1}
\times {\mathbb{Z}}_{m_2}\times \cdots \times {\mathbb{Z}}_{m_k}, $$ where $1 < m_1
\,|\,m_2 \,|\,\cdots \,|\,m_k=m$ and $m = s_i m_i$,
for all $i=1,\ldots,k$.
After fixing a generator $a_i\in {\mathbb{Z}}_{m_i}$
in each component and elements $\bar{s_i}\in {\mathbb{Z}}_{m}$ of order
$m_i$, any element $u=u_1a_1 +u_2a_2+\cdots +u_ka_k$
in $G$ is expressed as $u=(u_1,u_2,\cdots,u_k)\in G$ in this fixed
generators system.

The inner product of elements $u=(u_1,u_2,\cdots,u_k),
v=(v_1,v_2,\cdots,v_k)\in G$ is  defined uniquely by
\begin{equation}\label{innsubf}\langle u,v\rangle_m = \sum_{i} \bar{s_i}
\langle u_i,v_i\rangle_{m_i}  =\sum_{i}\bar{s_i}
\varphi_{v_i}(u_i)=\sum_{i}\bar{s_i} v_i u_i\in {\mathbb{Z}}_m.\end{equation} \end{prop}

\bigskip
Now, consider the finite abelian group ${\mathbb{Z}}_2^\alpha \times
{\mathbb{Z}}_4^\beta$ whose elements are vectors of $\alpha+\beta$ coordinates
(the first $\alpha$ over ${\mathbb{Z}}_2$ and the last $\beta$ over ${\mathbb{Z}}_4$).
By Proposition~\ref{defi}, fixing generators $a_i=1\in {\mathbb{Z}}_2$, for
$1\leq i\leq \alpha$, and $a_i\in\{1,3\}\in {\mathbb{Z}}_4$,  for
$\alpha+1\leq i\leq \alpha+\beta$, and also fixing the values
$\bar{s_i}=2$, for $1\leq i\leq \alpha$, which is the only possible
value of order two in ${\mathbb{Z}}_4$, and $\bar{s_i}=1\in\{1,3\}\subset
{\mathbb{Z}}_4$, for $\alpha+1\leq i\leq \alpha+\beta$, we can write the inner
product given by (\ref{innsubf}) in the following way that we will
call \textit{standard inner product}:

$$ \langle u,v \rangle=2(\sum_{i=1}^{\alpha} u_iv_i)+\sum_{j=\alpha+1}^{\alpha+\beta}
u_jv_j\in {\mathbb{Z}}_4,$$ where $u,v\in {\mathbb{Z}}_2^{\alpha}\times {\mathbb{Z}}_4^{\beta}$.

Note that although  $\bar{s_i}$ is uniquely defined for $1\leq i\leq
\alpha$, the value of $\bar{s_i}$, for $\alpha+1\leq i\leq
\alpha+\beta$, can be chosen from $\{1,3\}$ and so, we can produce
several different presentations for the inner product. Also note
that all of these different presentations of the inner product can
be reduced to the standard one, as long as in the computation of
$\langle u,v \rangle$ we take the representation of vector $u$ using
the given generators $a_i$ and the representation of vector $v$
using the generators $a'_i=a_i\in {\mathbb{Z}}_2$, for $1\leq i\leq \alpha$,
and $a_i'=\bar{s_i}a_i\in {\mathbb{Z}}_4$,  for $\alpha+1\leq i\leq
\alpha+\beta$.

We can also write the standard inner product as $$\langle u,v \rangle= u {\cdot} J_n {\cdot}
v^{t},$$ where
$\displaystyle J_n=\left (\begin{array}{c|c} 2I_{\alpha}& {\mathbf{0}} \\
\hline {\mathbf{0}} &I_{\beta}\end{array}\right )$ is a diagonal
matrix over ${\mathbb{Z}}_4$.
Note that when $\alpha=0$ the inner product is the usual one for
${\mathbb{Z}}_4$-vectors (i.e. vectors over ${\mathbb{Z}}_4$) and when $\beta=0$ it is
twice the usual one for ${\mathbb{Z}}_2$-vectors.

\bigskip
Let ${\cal C}$ be a $\Z_2\Z_4$-additive code of type
$(\alpha,\beta;\gamma,\delta;\kappa$) and let $C=\Phi({\cal C})$ be the
corresponding $\Z_2\Z_4$-linear code. The {\it additive orthogonal code}
of ${\cal C}$, denoted by ${\cal C}^\perp$, is defined in the standard
way $${\cal C}^\perp=\{v\in {\mathbb{Z}}_2^\alpha \times {\mathbb{Z}}_4^\beta \;|\;
\langle u,v \rangle =0 \mbox{ for all } u\in {\cal C}\}.$$ We will
also call ${\cal C}^\perp$ the {\it additive dual code} of ${\cal
C}$. The corresponding binary code $\Phi({\cal C}^\perp)$ is denoted
by $C_\perp$ and called {\it $\Z_2\Z_4$-dual code} of $C$. In the case
that $\alpha=0$, so when ${\cal C}$ is a quaternary linear code, ${\cal
C}^\perp$ is also called the {\it quaternary dual code} of ${\cal
C}$ and $C_\perp$ the {\it ${\mathbb{Z}}_4$-dual code} of $C$.

The additive dual code $\mathcal{C}^\perp$ is also a
$\Z_2\Z_4$-additive code, that is a subgroup of ${\mathbb{Z}}_2^{\alpha}\times
{\mathbb{Z}}_4^{\beta}$. Its weight enumerator polynomial is related to the
weight enumerator polynomial of $\mathcal{C}$ by McWilliams
Identity (see \cite{del}).  Notice that $C$ and $C_\perp$ are not
dual in the binary linear sense but the weight enumerator
polynomial of $C_\perp$ is the McWilliams transform of the weight
enumerator polynomial of $C$ (see \cite{del}, \cite{PropIT}).

\begin{lemm}\label{mcw} Let ${\cal C}$ be a $\Z_2\Z_4$-additive code of type
  $(\alpha,\beta;\gamma,\delta;\kappa)$ and ${\cal C}^\perp$ its additive dual code. Then,   $|\mathcal{C}||\mathcal{C}^\perp|=2^n$, where $n=\alpha+2\beta$.
\end{lemm}

\begin{demo} ~From the McWilliams Identity, $$
    W_{\mathcal{C}^\perp}(X,Y)=\frac{1}{|\mathcal{C}|}W_{\mathcal{C}}(X+Y,X-Y).
$$ Taking $X=Y$ we obtain, $$
    |\mathcal{C}^\perp|X^n=\frac{1}{|\mathcal{C}|}(2X)^{n-wt({\mathbf{0}})}
$$ and hence $|\mathcal{C}^\perp||\mathcal{C}|=2^n$. \end{demo}

Finally, notice again that one could think on $\Z_2\Z_4$-additive codes (or
$\Z_2\Z_4$-linear codes) only as quaternary linear codes (or
${\mathbb{Z}}_4$-linear codes), changing ones by twos in the coordinates
over ${\mathbb{Z}}_2$. However, they are not equivalent to the quaternary linear code, since the inner product
defined in ${\mathbb{Z}}_2^\alpha \times {\mathbb{Z}}_4^\beta$ gives us that the dual code of a $\Z_2\Z_4$-additive code is not
equivalent to the dual code of the corresponding quaternary linear code. Take, for example,
$\alpha=\beta=1$ and the vectors $v=(1,3)$ and $w=(1,2)$. It is
easy to check that $ \langle v,w \rangle =0$, so $v$ and $w$ are
orthogonal. If we change the ones by twos in the
coordinates over ${\mathbb{Z}}_2$ of these vectors we get $v'=(2,3)$ and
$w'=(2,2)$, which are not orthogonal in the quaternary sense.

\section{Parity-check matrices of $\Z_2\Z_4$-additive codes}
\label{sec:parityCheck}

In this section, first we will prove two different methods to
construct the additive dual code of a $\Z_2\Z_4$-additive code and we
will compute the type of this additive dual code. Then, we will
apply one of these two methods to show how to construct a
parity-check matrix of a $\Z_2\Z_4$-additive code, or equivalently a
generator matrix of its additive dual code, when the $\Z_2\Z_4$-additive
code is generated by a canonical generator matrix as in
(\ref{eq:StandardForm}).

\bigskip
Let ${\cal C}$ be a $\Z_2\Z_4$-additive code of type $(\alpha,\beta;\gamma,\delta;\kappa)$.
Since ${\cal C}$ is a subgroup of ${\mathbb{Z}}_2^{\alpha}\times {\mathbb{Z}}_4^{\beta}$,
the code ${\cal C}$ could be seen as the kernel of a group
homomorphism onto ${\mathbb{Z}}_2^{\bar{\gamma}}\times {\mathbb{Z}}_4^{\bar{\delta}}$, that is,
${\cal C}=ker \:\vartheta$, where $$ \vartheta:\;{\mathbb{Z}}_2^{\alpha}\times
{\mathbb{Z}}_4^{\beta} \longrightarrow\;{\mathbb{Z}}_2^{\bar{\gamma}}\times {\mathbb{Z}}_4^{\bar{\delta}}.
$$ The additive dual code ${\cal C}^\perp$ is also the kernel of
another group homomorphism onto ${\mathbb{Z}}_2^{\gamma}\times
{\mathbb{Z}}_4^{\delta}$, that is, ${\cal C}^\perp=ker \:{\bar \vartheta}$, where $$
{\bar \vartheta}:\;{\mathbb{Z}}_2^{\alpha}\times {\mathbb{Z}}_4^{\beta}
\longrightarrow\;{\mathbb{Z}}_2^{\gamma}\times {\mathbb{Z}}_4^{\delta}. $$

The homomorphism $\vartheta$ can be represented by a matrix $\cal{H}$,
which can be viewed as a parity-check matrix for the
$\Z_2\Z_4$-additive code ${\cal C}$ or as a generator matrix for its
additive dual code ${\cal C}^\perp$. Vice versa, the homomorphism
$\bar \vartheta$ can be represented by a matrix ${\cal G}$, which can be
viewed as a parity-check matrix for the additive dual code ${\cal
C}^\perp$ or as a generator matrix for the $\Z_2\Z_4$-additive code
${\cal C}$.

\begin{ex} \label{example:C1b} The code ${\cal C}_1$ (or the corresponding
$C_1=\Phi({\cal C}_1)$) in Example \ref{example:C1} is a $\Z_2\Z_4$-additive code
(or a $\Z_2\Z_4$-linear code) of type $(1,3;1,2;1)$ with generator
matrix
$${\cal G}_1= \left (\begin{array}{c|c c c} 1 & 2 & 2 & 2\\ \hline
                                     0 & 1 & 1 & 0\\
                                     1 & 1 & 2 & 3
\end{array}\right ).$$
The generator matrix ${\cal G}_1$ for ${\cal C}_1$ can be also viewed as a
parity-check matrix for its additive dual code ${\cal C}_1^\perp$.
Notice also that $|{\cal C}_1|=|C_1|=2\cdot 4^2=32$, so by Lemma
\ref{mcw}, $|{\cal C}_1^\perp|=2^7/32=4$. \end{ex}

\medskip
In order to construct the additive dual code of a $\Z_2\Z_4$-additive code,
we will need the following maps:
$\xi$ from ${\mathbb{Z}}_4$ to ${\mathbb{Z}}_2$ which is
the usual one modulo two, that is $\xi(0)=0$, $\xi(1)=1$, $\xi(2)=0$,
$\xi(3)=1$;
and the identity map $\iota$ from ${\mathbb{Z}}_2$ to ${\mathbb{Z}}_4$, that is
$\iota(0)=0$, $\iota(1)=1$. These maps can be extended to the maps
$(\xi, Id): {\mathbb{Z}}_4^\alpha\times{\mathbb{Z}}_4^\beta \longrightarrow
{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ and $(\iota,Id):
{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta \longrightarrow
{\mathbb{Z}}_4^\alpha\times{\mathbb{Z}}_4^\beta$, which will also be denoted by $\xi$
and $\iota$, respectively. Recall also the map $\chi$ from ${\mathbb{Z}}_2$ to
${\mathbb{Z}}_4$ which is the normal inclusion from the additive structure in
${\mathbb{Z}}_2$ to ${\mathbb{Z}}_4$, that is $\chi(0)=0$, $\chi(1)=2$; and its
extension  $(\chi,Id): {\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta \longrightarrow
{\mathbb{Z}}_4^\alpha\times{\mathbb{Z}}_4^\beta$, denoted also by $\chi$. We denote by
$\langle\cdot,\cdot\rangle_4$ the standard inner product for
quaternary vectors.

\medskip
\begin{lemm} \label{lemma:innerProduct}
If $u\in{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$, $v\in{\mathbb{Z}}_4^{\alpha+\beta}$, then
$\langle \chi(u),v \rangle_4=\langle u,\xi(v)\rangle.$
\end{lemm}

\begin{demo}
$\langle \chi(u),v\rangle_4 = \sum_{i=1}^{\alpha}
(2u_i)v_i+\sum_{j=\alpha+1}^{\alpha+\beta}u_jv_j=\sum_{i=1}^{\alpha}
(2u_i)(v_i \bmod 2)+\sum_{j=\alpha+1}^{\alpha+\beta}u_jv_j= \langle
u,\xi(v) \rangle$.
\end{demo}

\begin{coro}\label{coro:chiiota}
If $u,v\in{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$, then $
\langle \chi(u),\iota(v) \rangle_4=\langle u,v\rangle.$
\end{coro}

\begin{demo}
By Lemma \ref{lemma:innerProduct}, $\langle \chi(u),\iota(v)
\rangle_4=\langle u,\xi(\iota(v))\rangle=\langle u,v\rangle$.
\end{demo}

\begin{prop}\label{prop:xichiperp} Let $\cal{C}$ be a $\Z_2\Z_4$-additive
code of type $(\alpha,\beta;\gamma,\delta;\kappa)$. Then, $$
\cal{C}^\perp=\xi(\chi(\cal{C})^\perp).$$ \end{prop}

\begin{demo}
We know that if $v\in\cal{C}^\perp$, then $\langle u,v \rangle=0$, for all $u\in\cal{C}$.
By Corollary \ref{coro:chiiota}, $\langle u,v \rangle=\langle \chi(u), \iota(v) \rangle _4=0$.
Therefore, $\xi(\iota(v))=v \in \xi(\chi(\cal{C})^\perp)$ and $\cal{C}^\perp \subseteq \xi(\chi(\cal{C})^\perp)$.
On the other hand, if $v\in \chi({\cal C})^\perp$, then $\langle \chi(u),v \rangle _4=0$, for all $u\in\cal{C}$.
By Lemma \ref{lemma:innerProduct}, $\langle \chi(u),v \rangle _4=\langle u,\xi(v)\rangle =0$.
Thus, $\xi(\chi(\cal{C})^\perp)\subseteq \cal{C}^\perp$ and we
obtain the equality.
\end{demo}

\begin{prop}\label{prop:xihperp} Let $\cal{C}$ be a $\Z_2\Z_4$-additive code
of type $(\alpha,\beta;\gamma,\delta;\kappa)$.
Then, $$\mathcal{C}^\perp=\chi^{-1}(\xi^{-1}(\mathcal{C})^\perp).$$
\end{prop}

\begin{demo} Let ${\cal G}$ be a generator matrix of the $\Z_2\Z_4$-additive code
$\cal{C}$ written as in (\ref{eq:matrixG}). Then,
the quaternary linear code $\xi^{-1}({\cal C})$ has a generator matrix
of the form \begin{equation} \label{ximat} \left (\begin{array}{c c}
                2I_{\alpha} & {\mathbf{0}} \\
                B_1 & 2B_3 \\
                B_2 & Q \\
 \end{array}\right ).\end{equation}
 We will show that $v \in \mathcal{C}^\perp$ if and only if $ \chi(v)\in \xi^{-1}({\cal C})^\perp$. In fact,
for each row vector $f$ in the matrix $(2I_{\alpha} \ {\mathbf{0}})$, we
have $\langle \chi(v),f \rangle _4= \sum_{i=1}^\alpha f_i 2v_i=0$
because there is only one index $i$ such that $f_i=2$. Moreover, by
Corollary \ref{coro:chiiota}, $0=\langle v,u\rangle=\langle
\chi(v),\iota(u) \rangle_4$, for all $u \in \mathcal{C}$.
\end{demo}

The following question we will settle is the computation of the type
of the additive dual code of a given $\Z_2\Z_4$-additive code
${\cal C}$. First, we will remember
this well-known result for quaternary linear codes, that is for $\Z_2\Z_4$-additive
codes with $\alpha=0$. Then, we will generalize it  for
$\Z_2\Z_4$-additive codes, not necessarily quaternary linear codes.

\begin{lemm}{\cite{Sole}} \label{z4}
If ${\cal C}$ is a quaternary linear code of type $(0,\beta;\gamma,\delta;0)$,
then the quaternary dual code ${\cal C}^\perp$ is of type
$(0,\beta;\gamma,\beta-\gamma-\delta;0)$.
\end{lemm}

\begin{theo} \label{parameters} Let
${\cal C}$ be a $\Z_2\Z_4$-additive code of type
$(\alpha,\beta;\gamma,\delta;\kappa)$. The additive dual code
${\cal C}^\perp$ is then of type $(\alpha,\beta;\bar{\gamma},\bar{\delta};\bar{\kappa})$,
where $$\begin{array}{l} \bar{\gamma} = \alpha + \gamma - 2\kappa,\\
\bar{\delta} =\beta - \gamma - \delta + \kappa,\\
 \bar{\kappa}=\alpha-\kappa. \end{array}$$
\end{theo}

\begin{demo} Let ${\cal G}$ be a generator matrix of the $\Z_2\Z_4$-additive code
$\cal{C}$ written as in (\ref{eq:matrixG}). Then, the matrix (\ref{ximat}) is a
generator matrix for the quaternary linear code
$\xi^{-1}({\cal C})$, which is of type $(0,\alpha+\beta; \gamma',
\delta';0)$, where $\gamma'=\alpha+\gamma-2\kappa$ and
$\delta'=\delta+\kappa$. The value of $\delta'$ comes from the
fact that the $\kappa$ independent binary vectors of $({\cal C}_b)_X$ are in
$A$ and, so, the number of independent quaternary vectors of order
four becomes $\delta+\kappa$. The value of $\gamma'$ comes from
the fact that the cardinality of the quaternary linear code
$\xi^{-1}({\cal C})$ is
$2^{\gamma'+2\delta'}=2^{\gamma+2\delta+\alpha}$.

By Lemma \ref{z4},
the quaternary dual code $\xi^{-1}({\cal C})^\perp$  is of type $(0,\alpha+\beta; \bar{\gamma}, \bar{\delta};0)$,
where $\bar{\gamma}=\gamma'$ and $\bar{\delta}=\alpha+\beta-\gamma'-\delta'
=\alpha+\beta-(\gamma+\alpha-2\kappa)-(\delta+\kappa)=
\beta-\gamma-\delta+\kappa$.

Note that the $\bar{\delta}$ independent vectors in
$\xi^{-1}({\cal C})^\perp$, restricted to the first $\alpha$ coordinates, are
vectors of order two, because in $\xi^{-1}({\cal C})$ there are the
row vectors of the matrix $(2I_{\alpha} \ {\mathbf{0}})$. Finally, applying $\chi^{-1}$ we
obtain the additive dual code of ${\cal C}$. For this additive dual code ${\cal C}^\perp$,
the value of $\bar{\kappa}$ can be easily computed from the fact that,
again, the additive dual coincides with ${\cal C}$.
\end{demo}

There are two different methods to obtain the additive dual code
${\cal C}^\perp$, one given by Proposition \ref{prop:xichiperp} and
another one by Proposition \ref{prop:xihperp}.  Using any of these
two methods, we can construct a generator matrix of ${\cal C}^\perp$,
or equivalently a parity-check matrix of ${\cal C}$, starting from a
generator matrix of ${\cal C}$. In Example \ref{example:C1checkmat}, we
consider the canonical generator matrix of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive
code and apply these two methods to obtain a generator matrix of its
additive dual code. Note that the process to obtain this matrix is
different using both methods but, in this case, the generator
matrices obtained coincide.

Theorem \ref{prop:Dual1} shows how to construct the parity-check matrix of a $\Z_2\Z_4$-additive code
generated by a canonical generator matrix as in (\ref{eq:StandardForm}). This result is proved using the method given by Proposition \ref{prop:xichiperp}.
Notice also that we can apply any of the two methods to any generator matrix, not necessary a
canonical generator matrix, to get a parity-check matrix.

\begin{lemm} \cite{Sole} \label{lem:QaryDual1} If $\mathcal{C}$ is a quaternary linear code of type
$(0,\beta;\gamma,\delta;0)$ with canonical generator matrix
(\ref{eq:QaryStandardForm}), then the generator matrix of
$\mathcal{C}^\perp$ is  \begin{equation}  \cal{H}_S=\left (
\begin{array}{|ccc}
{\mathbf{0}}                   & 2I_{\gamma}& 2R^{t}\\ \hline
I_{\beta-\gamma-\delta} & T^{t}     & -(S+RT)^{t}\\
\end{array} \right ),\end{equation} where $R,T$ are
matrices over ${\mathbb{Z}}_2$ of size $\delta\times\gamma$ and
$\gamma\times(\beta-\gamma-\delta)$, respectively; and $S$ is a
matrix over ${\mathbb{Z}}_4$ of size $\delta\times(\beta-\gamma-\delta)$.
\end{lemm}

\begin{theo} \label{prop:Dual1} Let $\mathcal{C}$ be a $\Z_2\Z_4$-additive code of type
$(\alpha,\beta;\gamma,\delta;\kappa)$ with canonical generator matrix
(\ref{eq:StandardForm}). Then, the generator matrix of
$\mathcal{C}^\perp$ is  \begin{equation}  \cal{H}_S=\left (
\begin{array}{cc|ccc}
T_b^{t} & I_{\alpha-\kappa} & {\mathbf{0}} &  {\mathbf{0}} & 2S_b^{t}\\
{\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & 2I_{\gamma-\kappa}& 2R^{t}\\
\hline T_2^{t} &{\mathbf{0}} & I_{\beta+\kappa-\gamma-\delta} &  T_1^{t}
& -\big( S_q+RT_1\big)^{t} \end{array} \right ),\end{equation}
where $T_b, T_1, T_2, R, S_b$ are matrices over ${\mathbb{Z}}_2$
and $S_q$ is a matrix over ${\mathbb{Z}}_4$.
\end{theo}

\begin{demo} By Lemma \ref{lem:QaryDual1}, if $\bar{{\cal C}}$ is a quaternary linear code
with generator matrix $$\bar{{\cal G}}= \left ( \begin{array}{|ccccc}
2T_b & 2T_2 & 2I_{\kappa} & {\mathbf{0}} & {\mathbf{0}}\\
{\mathbf{0}} & 2T_1 & {\mathbf{0}} & 2I_{\gamma-\kappa} & {\mathbf{0}}\\ \hline
2S_b & S_q & {\mathbf{0}} & R & I_{\delta} \end{array} \right ),$$ then
the quaternary dual code $\bar{{\cal C}}^\perp$ has generator matrix $\bar{\cal{H}}=$
$$ \left ( \begin{array}{|ccccc}
{\mathbf{0}} & {\mathbf{0}} & 2I_{\kappa} & {\mathbf{0}} & {\mathbf{0}} \\
{\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & 2I_{\gamma-\kappa}& 2R^{t}\\ \hline
I_{\alpha-\kappa} & {\mathbf{0}} & T_b^{t} & {\mathbf{0}} & 2S_b^{t}\\
{\mathbf{0}} & I_{\beta-\gamma-\delta+\kappa} & T_2^{t} & T_1^{t} &
-\big( S_q+RT_1\big)^{t} \end{array} \right ). $$
Hence, if ${\cal C}$ is a $\Z_2\Z_4$-additive code with generator
matrix (\ref{eq:StandardForm}), then the generator matrix of
$\chi({\cal C})^\perp$ is $\cal{H}_\xi=$  $$\left (
\begin{array}{|ccccc}
2I_{\kappa} &{\mathbf{0}} & {\mathbf{0}} &  {\mathbf{0}} & {\mathbf{0}} \\
{\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & 2I_{\gamma-\kappa}& 2R^{t}\\ \hline
T_b^{t} & I_{\alpha-\kappa} & {\mathbf{0}} &  {\mathbf{0}} & 2S_b^{t}\\
T_2^{t} &{\mathbf{0}} & I_{\beta-\gamma-\delta+\kappa} &  T_1^{t} &
-\big( S_q+RT_1\big)^{t} \end{array} \right ). $$
Finally, by Proposition \ref{prop:xichiperp}, $\cal{H}_S=\xi(\cal{H}_\xi)$ is
the generator matrix of ${\cal C}^\perp$. \end{demo}

Note that by Theorem \ref{parameters} and Theorem \ref{prop:Dual1},
if $\mathcal{C}$ is a $\Z_2\Z_4$-additive code of type
$(\alpha,\beta;\gamma,\delta;\kappa)$ with canonical generator matrix
(\ref{eq:StandardForm}), then ${\cal C}^\perp$ is permutation equivalent
to a $\Z_2\Z_4$-additive code with canonical generator matrix
\begin{equation}  \left ( \begin{array}{cc|ccc}
I_{\bar{\kappa}} & T_b^{t} & 2S_b^{t} & {\mathbf{0}} &  {\mathbf{0}} \\
{\mathbf{0}} & {\mathbf{0}} & 2R^{t} & 2I_{\bar{\gamma}-\bar{\kappa}}& {\mathbf{0}} \\ \hline
 {\mathbf{0}} & T_2^{t} & -\big( S_q+RT_1\big)^{t} &  T_1^{t} &
I_{\bar{\delta}} \end{array} \right ), \end{equation} where $T_b, T_1, T_2, R, S_b$ are matrices over ${\mathbb{Z}}_2$;
$S_q$ is a matrix over ${\mathbb{Z}}_4$, $\bar{\gamma}=\alpha+\gamma-2\kappa$,
$\bar{\delta}=\beta-\gamma-\delta+\kappa$ and $\bar{\kappa}=\alpha-\kappa$.

\medskip
\begin{ex} \label{example:C1checkmat}
Let ${{\cal C}_{S1}}$ denote the $\Z_2\Z_4$-additive code of type $(1,3;1,2;1)$ with canonical
generator matrix
$${\cal G}_S=\left (\begin{array}{c|ccc} 1 & 2 & 0 & 0\\ \hline
                                     0 & 1 & 1 & 0\\
                                     0 & 3 & 0 & 1
\end{array}\right ).$$
By Theorem \ref{parameters}, the additive dual code ${\cal C}_{S1}^\perp$ is of type $(1,3;0,1;0)$.
There are two methods to obtain a parity-check matrix of ${{\cal C}_{S1}}$ from the matrix
${\cal G}_S$.

The first one uses Proposition \ref{prop:xichiperp}.
We know that if $\bar{\cal C}$ is a quaternary linear code with generator matrix
$\bar{{\cal G}} =\left (\begin{array}{|cccc} 2 & 2 & 0 & 0\\ \hline
                            1 & 0 & 1 & 0\\
                            3 & 0 & 0 & 1
\end{array}\right ),$ the quaternary dual code $\bar{\cal C}^\perp$ has generator matrix
$\bar\cal{H} =\left (\begin{array}{|c ccc} 0 & 2 & 0 & 0\\  \hline
                            1 & 1 & 3 & 1 \\
\end{array}\right ).$ So, the generator matrix of $\chi({\cal C}_{S1})^\perp$ is
$\left (\begin{array}{|c ccc} 2 & 0 & 0 & 0\\ \hline
                            1 & 1 & 3 & 1 \\
\end{array}\right )$
and finally, applying $\xi$, the generator matrix of
${\cal C}_{S1}^\perp=\xi(\chi({\cal C}_{S1})^\perp)$
is $${\cal{H}_S} =\left (\begin{array}{c|ccc}  1 & 1 & 3 & 1\\
\end{array}\right ).$$

The second method uses Proposition \ref{prop:xihperp}. We know that the quaternary
linear code $\xi^{-1}({\cal C}_{S1})$ with generator matrix
$$\left (\begin{array}{|c c c c} 2 & 0 & 0 & 0 \\ \hline
                                  1 & 2 & 0 & 0 \\
                                  0 & 1 & 1 & 0 \\
                                  0 & 3 & 0 & 1 \\
                                   \end{array}\right ),$$
or equivalently $\left (\begin{array}{|c c c c} \hline
                                  1 & 2 & 0 & 0 \\
                                  0 & 1 & 1 & 0 \\
                                  0 & 3 & 0 & 1 \\
                                   \end{array}\right )$,
has parity-check matrix $\left (\begin{array}{|cccc} \hline  2 & 1 & 3 & 1\\
\end{array}\right ).$ So, applying $\chi^{-1}$, the generator matrix of
${\cal C}_{S1}^\perp=\chi^{-1}(\xi^{-1}({\cal C}_{S1})^\perp)$
is $${\cal{H}_S} =\left (\begin{array}{c|ccc}  1 & 1 & 3 & 1\\
\end{array}\right ).$$
\end{ex}

\begin{ex} Let ${{\cal C}_{S2}}$ be a $\Z_2\Z_4$-additive code of type $(3,4;3,1;1)$ with canonical generator
matrix
$$\left (\begin{array}{ccc|cccc}1 & 0 & 0  &  2 & 2 & 0 & 0 \\
                                0 & 1 & 0  &  0 & 0 & 0 & 0 \\
                                0 & 0 & 1  &  2 & 2 & 0 & 0 \\\hline
                                0 & 0 & 0  &  1 & 1 & 1 & 1 \\
\end{array}\right ).$$
By Theorem \ref{parameters} and Theorem \ref{prop:Dual1}, the additive dual
code ${\cal C}_{S2}^\perp$ is of type $(3,4;0,3;0)$ and has generator matrix
$$\left (\begin{array}{ccc|cccc}\hline 1 & 0 & 1  &  1 & 0 & 0 & 3 \\
                                       1 & 0 & 1  &  0 & 1 & 0 & 3 \\
                                       0 & 0 & 0  &  0 & 0 & 1 & 3 \\
\end{array}\right ).$$
\end{ex}

\section{Additive self-dual codes}
\label{sec:selfdual}

Let ${\cal C}$ be a $\Z_2\Z_4$-additive code. We say that ${\cal C}$ is an {\it additive self-orthogonal code} if ${\cal C}\subseteq {\cal C}^\perp$
and ${\cal C}$ is an {\it additive self-dual code} if ${\cal C}= {\cal C}^\perp$.
Let $C=\Phi({\cal C})$ be the corresponding
$\Z_2\Z_4$-linear code. We say that $C$ is a {\em self\/}
$\Z_2\Z_4$-{\em orthogonal code} if $C\subseteq C_\perp$ and $C$ is a {\em
self\/} $\Z_2\Z_4$-{\em dual code} if $C=C_\perp$, where $C_\perp=\Phi({\cal C}^\perp)$.
In this section, we will study the additive self-dual codes.

\bigskip
Note that in the case that $\beta=0$, that is when ${\cal C}=C$ is a binary
linear code, we will also say that ${\cal C}$ is {\em binary
self-orthogonal} (or {\em binary self-dual}) if ${\cal C} \subseteq
{\cal C}^\perp$ (or ${\cal C}= {\cal C}^\perp$). And in the case that
$\alpha=0$, that is when ${\cal C}$ is a quaternary linear code, we will
also say that ${\cal C}$ is {\em quaternary self-orthogonal} (or {\em
quaternary self-dual}) if ${\cal C} \subseteq {\cal C}^\perp$ (or ${\cal C}=
{\cal C}^\perp$).

Recall that ${\cal C}_X$ is the punctured code of ${\cal C}$ by deleting
the coordinates outside $X$, ${\cal C}_Y$ is the punctured code of
${\cal C}$ by deleting the coordinates outside $Y$ and ${\cal C}_b$ is the
subcode of ${\cal C}$ which contains all codewords of order two.
Denote by $w(u)$ the Hamming weight of any vector $u\in {\mathbb{Z}}_2^\alpha$.

\bigskip
\begin{lemm}\label{sd:parameters}
If ${\cal C}$ is an additive self-dual code, then ${\cal C}$ is of type
$(2\kappa,\beta;\beta+\kappa-2\delta,\delta;\kappa)$,
$|{\cal C}|=2^{\kappa+\beta}$ and $|{\cal C}_b|=2^{\kappa+\beta-\delta}$.
\end{lemm}

\begin{demo}
By Theorem \ref{parameters}, we have that $\alpha=2\kappa$ and
$\gamma=\beta+\kappa-2\delta$. Since $|{\cal C}|=2^{\gamma+2\delta}$ and
$|{\cal C}_b|=2^{\gamma+\delta}$, the result holds.
\end{demo}

\begin{lemm}\label{sd:evenodd}
Let ${\cal C}$ be an additive self-dual code and let $z=(x\mid y)\in
{\cal C}$. Denote by $p(u)$ the number of odd (order four) coordinates
of any vector $u\in {\mathbb{Z}}_4^\beta$. Then,
\begin{itemize}
\item[(i)] if $w(x)$ is even, then $p(y)\equiv 0 \pmod 4$.
\item[(ii)] if $w(x)$ is odd, then $p(y)\equiv 2 \pmod 4$.
\item[(iii)] $({\mathbf{0}}\mid \mathbf{2})$ is a codeword in ${\cal C}$.
\end{itemize}
\end{lemm}

\begin{demo}
$(i)$ and $(ii)$ follows easily since $z$ must be orthogonal to
itself and we have $\langle z,z \rangle=2w(x)+p(y)=0 \in {\mathbb{Z}}_4$. Now,
$(iii)$ is obvious because $p(y)$ is always even.
\end{demo}

\begin{lemm}\label{sd:binariautodual}
If ${\cal C}$ is an additive self-dual code,
then the subcode $({\cal C}_b)_X$ is a binary self-dual code.
\end{lemm}

\begin{demo}
By Lemma \ref{sd:parameters}, the code ${\cal C}$ is of type $(2\kappa,\beta;\beta+\kappa-2\delta,\delta;\kappa)$.
Since for any pair of codewords $(x\mid y), (x'\mid
y')\in {\cal C}_b$ we have $\langle y,y' \rangle_4=0$, $({\cal C}_b)_X\subseteq ({\cal C}_b)^\perp_X$.
Moreover, since $({\cal C}_b)_X$ has dimension $\kappa$
(by definition) and is of length $2\kappa$, we have that $({\cal C}_b)_X$ is
binary self-dual.
\end{demo}

\begin{lemm}\label{sd:cardinalquaternari}
Let ${\cal C}$ be an additive self-dual code of type $(2\kappa,\beta; \beta+\kappa-2\delta,\delta;\kappa)$.
There is an integer number $r$, $0\leq r\leq \kappa$, such that each
codeword in ${\cal C}_Y$ appears $2^r$ times in ${\cal C}$ and
$|{\cal C}_Y|\geq 2^\beta$.
\end{lemm}

\begin{demo}
Consider the subcode ${\cal C}_0=\{(x\mid {\mathbf{0}})\in {\cal C}\}$. Clearly,
$({\cal C}_0)_X$ is a binary linear code. Let $r=dim({\cal C}_0)_X$. Thus,
any vector in ${\cal C}_Y$ appears $2^r$ times in ${\cal C}$. Note that
$({\cal C}_0)_X$ is also a subcode of $({\cal C}_b)_X$, hence $r\leq \kappa$.
Also, we have that $|{\cal C}|=2^{\beta +\kappa}=|{\cal C}_Y|\cdot 2^r$,
therefore $|{\cal C}_Y|\geq 2^\beta$.
\end{demo}

We say that a binary code $C$ is {\it antipodal} if for any codeword
$z\in C$, $z+\u \in C$. The following two examples show us two
different cases of additive self-dual codes. In Example
\ref{sd:antipodal}, the corresponding $\Z_2\Z_4$-linear code
$C_1=\Phi({\cal C}_1)$ is antipodal, or equivalently ${\cal C}_1$ contains
the codeword $(\u\mid \mathbf{2})$. On the other hand, in Example
\ref{sd:noantipodal}, the corresponding $\Z_2\Z_4$-linear code
$C_2=\Phi({\cal C}_2)$ is not antipodal. We will study these two cases
separately.

\begin{ex}\label{sd:antipodal}
An additive self-dual code with $\alpha,\beta \geq  1$ should have
$\alpha\geq 2$, since $\alpha$ must be even. An additive self-dual
code with minimum number of coordinates has $\alpha=2$, $\beta=1$
and $2^{\kappa+\beta}=2^{1+1}=4$ codewords. For example, the code
${\cal C}_1=\{(00\mid 0), (00\mid 2), (11\mid 0), (11\mid 2)\}$ is an
additive self-dual code of type $(2,1;2,0;1)$ and has generator
matrix
$${\cal G}_1=\left (\begin{array}{cc|c}    1  &  1 & 0 \\
                                0  &  0 & 2 \\\hline
\end{array}\right ).$$
\end{ex}

\begin{ex}\label{sd:noantipodal}
Let ${\cal C}_b=\{(00\mid 00), (00\mid 22), (11\mid 02), (11\mid 20)\}$.
Then, the code ${\cal C}_2={\cal C}_b \cup ({\cal C}_b+(01\mid 11))$ is an additive
self-dual code of type $(2,2;1,1;1)$ and has generator matrix
$${\cal G}_2=\left (\begin{array}{cc|cc}  1 & 1  &  2 & 0 \\ \hline
                                0  &1 &  1 & 1 \\
\end{array}\right ).$$
\end{ex}

The following Lemmas \ref{lemEx1} and \ref{lemEx2} give us two generalizations of Example \ref{sd:noantipodal}. Note that any of the corresponding $\Z_2\Z_4$-linear codes are not antipodal.

\begin{lemm} \label{lemEx1} If $\delta \leq \kappa$, the $\Z_2\Z_4$-additive code ${\cal C}$ of type $(2\kappa,\beta;\beta+\kappa-2\delta,\delta;\kappa)$
with canonical generator matrix $${\cal G}=\left (\begin{array}{cccc|ccc}
    I_\delta  & {\mathbf{0}} & I_\delta & {\mathbf{0}} & 2I_\delta & {\mathbf{0}} & {\mathbf{0}} \\
       {\mathbf{0}}  & I_{\kappa-\delta}& {\mathbf{0}} & I_{\kappa-\delta} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\
       {\mathbf{0}} & {\mathbf{0}} &  {\mathbf{0}}  &  {\mathbf{0}} & {\mathbf{0}} & 2I_{\beta-2\delta} & {\mathbf{0}} \\ \hline
       {\mathbf{0}}  & {\mathbf{0}} & I_\delta & {\mathbf{0}} & I_\delta & {\mathbf{0}} & I_\delta \\
\end{array}\right )$$ is an additive self-dual code.
\end{lemm}

\begin{demo} Straightforward using Theorem \ref{prop:Dual1}. \end{demo}

\begin{lemm} \label{lemEx2} If $\delta \leq \kappa$,
the $\Z_2\Z_4$-additive code ${\cal C}$ of type $(2\kappa,\beta;\beta+\kappa-2\delta,\delta;\kappa)$
with canonical generator matrix $${\cal G}=\left (\begin{array}{cccc|ccc}
    I_\delta  & {\mathbf{0}} & I_\delta & {\mathbf{0}} & 2I_\delta & {\mathbf{0}} & {\mathbf{0}} \\
       {\mathbf{0}}  & I_{\kappa-\delta}& {\mathbf{0}} & I_{\kappa-\delta} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\
       {\mathbf{0}} & {\mathbf{0}} &  {\mathbf{0}}  &  {\mathbf{0}} & {\mathbf{2}} & 2I_{\beta-2\delta} & {\mathbf{0}} \\ \hline
       {\mathbf{0}}  & {\mathbf{0}} & I_\delta & {\mathbf{0}} & I_\delta & \u & I_\delta \\
\end{array}\right )$$ is an additive self-dual code if and only if $\beta -2\delta \equiv 0 \pmod 4$.
\end{lemm}

\begin{demo} Let $e_i$ denote the vector  with all components equal to zero, except the
$i$th component, which contains a one. It is easy to  that any two
rows of the generator matrix ${\cal G}$ are orthogonal. Notice that the
rows of order four $u=({\mathbf{0}} \ {\mathbf{0}} \ e_i \ {\mathbf{0}} \mid e_i \ \u \
e_i)$  are orthogonal if and only if $\beta -2\kappa \equiv 0\pmod
4$. By Theorem \ref{prop:Dual1}, we have $|{\cal C}|=|{\cal C}^\perp|$, so
${\cal C}$ is additive self-dual.
\end{demo}

\begin{prop}\label{sd:propantipodal}
Let $C$ be a self $\;\Z_2\Z_4$-dual code of length $n$ and let $A_i$
denote the number of codewords of weight $i$ ($0\leq i\leq n$). The
following statements are equivalent:
\begin{itemize}
\item[(i)] $C$ is antipodal.
\item[(ii)] $C_X$ has only even weights (and also $C$).
\item[(iii)] $\sum_{i=0}^n (-1)^i A_i=|C|$.
\end{itemize}
\end{prop}

\begin{demo}
Let ${\cal C}$ be the corresponding $\Z_2\Z_4$-additive code ${\cal C}=\Phi^{-1}(C)$.
Note that given a codeword $z=(x\mid y)\in {\cal C}$, where
$y=(y_1,\ldots,y_\beta)$, the weight of $(\phi(y_1),\ldots,\phi(y_\beta))$ is always even.
Thus, the parity of the weight of $\Phi(z)$
is the same as the parity of the weight of $x$.

$(i)\;\Leftrightarrow\;(ii)$: If $C$ is antipodal, then $(\u \mid
\mathbf{2})\in {\cal C}$. Therefore, for any codeword $(x\mid y)\in
{\cal C}$, $w(x)$ must be even. Reciprocally, if $w(x)$ is even for any
codeword $(x\mid y)\in {\cal C}$, then the vector $(\u \mid
\mathbf{2})$ is orthogonal to any codeword and hence $(\u \mid
\mathbf{2})\in {\cal C}$.

$(ii)\;\Leftrightarrow\;(iii)$: Straightforward because
$\sum_{i=0}^n A_i=|C|$.
\end{demo}

\begin{prop}
Let $C$ be a self $\;\Z_2\Z_4$-dual code of length $n$ and let $A_i$ denote
the number of codewords of weight $i$ ($0\leq i\leq n$).
The following statements are equivalent:
\begin{itemize}
\item[(i)] $C$ is not antipodal.
\item[(ii)] $C_X$ has even and odd weights (and also $C$).
\item[(iii)] $\sum_{i=0}^n (-1)^i A_i=0$.
\end{itemize}
\end{prop}

\begin{demo}
Statements $(i)$ and $(ii)$ are equivalent by Proposition
\ref{sd:propantipodal}.

$(i)\;\Leftrightarrow\;(iii)$: Consider the MacWilliams Identity:
$$
W_C(X,Y)=\frac{1}{|C|} W_{C_\perp}(X+Y,X-Y),
$$
where $W_C(X,Y)$ is the weight enumerator polynomial of $C$:
$$
W_C(X,Y)=\sum_{i=0}^n A_i X^{n-i} Y^i.
$$
Since $C=C_\perp$ and taking $X=0$, we obtain:
$$
A_n Y^n = \frac{1}{|C|}\sum_{i=0}^n (-1)^i A_i
Y^n\;\;\Longrightarrow\;\;|C|A_n=\sum_{i=0}^n (-1)^i A_i.
$$
Finally, since $A_n=1$ when $C$ is antipodal and $A_n=0$ when $C$ is not
antipodal, we have that $\sum_{i=0}^n (-1)^i A_i=0$ if and only if
$C$ is not antipodal.
\end{demo}

\begin{prop} \label{prop:antipodal}
Let ${\cal C}$ be an additive self-dual code of
type $(2\kappa,\beta; \beta+\kappa-2\delta,\delta;\kappa)$. The following statements
are equivalent:
\begin{itemize}
\item[(i)] ${\cal C}_X$ is binary self-orthogonal.
\item[(ii)] ${\cal C}_X$ is binary self-dual.
\item[(iii)] $|{\cal C}_X|=2^\kappa$.
\item[(iv)] ${\cal C}_Y$ is a quaternary self-orthogonal code.
\item[(v)] ${\cal C}_Y$ is a quaternary self-dual code.
\item[(vi)] $|{\cal C}_Y|=2^\beta$.
\item[(vii)] ${\cal C} = {\cal C}_X\oplus{\cal C}_Y$.
\end{itemize}
\end{prop}

\begin{demo}
$(i)\;\Leftrightarrow\;(ii)$: By Lemma \ref{sd:binariautodual},
$|{\cal C}_X|\geq 2^\kappa$, thus $(i)$ and $(ii)$ are equivalent
statements.

$(ii)\;\Leftrightarrow\;(iii)$: Clearly, $(ii)$ implies $(iii)$ and
$(iii)$ implies ${\cal C}_X=({\cal C}_b)_X$ and  ${\cal C}_X$
is binary self-dual, by Lemma \ref{sd:binariautodual}.

$(ii)\;\Leftrightarrow\;(v)$: Straightforward.

$(iv)\;\Leftrightarrow\;(v)$: By Lemma \ref{sd:cardinalquaternari},
$|{\cal C}_Y|\geq 2^\beta$, thus $(iv)$ and $(v)$ are equivalent
statements.

$(ii)\;\Leftrightarrow\;(vii)$: If ${\cal C}_X$ is binary self-dual, then
${\cal C}_Y$ is quaternary self-dual, $|{\cal C}_X|=2^\kappa$ and
$|{\cal C}_Y|=2^\beta$. Since ${\cal C}=2^{\kappa+\beta}$ we have that the
set of codewords in ${\cal C}$ is ${\cal C}_X \times {\cal C}_Y$.
Reciprocally, if ${\cal C} = {\cal C}_X\oplus{\cal C}_Y$, then $(x\mid
{\mathbf{0}})\in {\cal C}$ for any $x\in{\cal C}_X$ and ${\cal C}_X$ must be a
binary self-dual code. Also, $({\mathbf{0}}\mid y)\in{\cal C}$ for any $y\in{\cal C}_Y$ and
${\cal C}_Y$ must be a quaternary self-dual code.

$(v)\;\Rightarrow\;(vi)$: Trivial.

$(vi)\;\Rightarrow\;(iii)$: By Lemma \ref{sd:cardinalquaternari},
each vector in ${\cal C}_Y$ appears $2^\kappa$ times in ${\cal C}$. Thus,
for any vector $x_b\in ({\cal C}_b)_X$, the vector $(x_b\mid{\mathbf{0}})$ is a codeword
in ${\cal C}_b$. This means that given any codeword $(x\mid y)\in{\cal C}$, we
have that $\langle x,x_b \rangle=0$, for all $x_b\in({\cal C}_b)_X$, since $ \langle (x\mid
y),(x_b,{\mathbf{0}})\rangle = 0$. Therefore, for all $x\in{\cal C}_X$, $x\in ({\cal C}_b)_X^\perp$
and ${\cal C}_X \subseteq ({\cal C}_b)_X^\perp$. By Lemma \ref{sd:binariautodual},
$({\cal C}_b)_X^\perp=({\cal C}_b)_X$, which implies ${\cal C}_X=({\cal C}_b)_X$,
hence $|{\cal C}_X|=2^\kappa$.
\end{demo}

It is easy to check that if ${\cal C}$ is an additive self-dual code, then the
codewords in ${\cal C}_X^\perp\oplus{\cal C}_Y^\perp$ are orthogonal to
${\cal C}$ and, hence, ${\cal C}_X^\perp\oplus{\cal C}_Y^\perp\subseteq
{\cal C}^\perp$.

\begin{prop}
  If ${\cal C}_X$ is a binary self-dual code of length $\alpha=2\kappa$
  and ${\cal C}_Y$ is a quaternary self-dual code of type
  $(0,\beta;\gamma,\delta;0)$, then
  ${\cal C}={\cal C}_X\oplus{\cal C}_Y$ is an additive self-dual code of type
  $(2\kappa,\beta;\beta+\kappa-2\delta,\delta;\kappa)$.
\end{prop}

\begin{demo} Since ${\cal C}_X$ is binary self-dual, $|{\cal C}_X|=2^\kappa$, where $\alpha=2\kappa$.
Since ${\cal C}_Y$ is quaternary self-dual, $|{\cal C}_Y|=2^\beta=2^{\gamma+2\delta}$,
  so $\gamma=\beta-2\delta$.
  Let ${\cal G}_X$ be a generator matrix of ${\cal C}_X$ of size $\kappa\times 2\kappa$
  and let ${\cal G}_Y$ be a  generator
  matrix of ${\cal C}_Y$ of size $(\beta-\delta) \times \beta$.  Then, the matrix ${\cal G}$ defined as
$${\cal G}=\left (\begin{array}{c|c} {\cal G}_X & {\mathbf{0}} \\
                            {\mathbf{0}} & {\cal G}_Y
\end{array}\right )$$
\noindent is a generator matrix of ${\cal C}={\cal C}_X\oplus{\cal C}_Y$. It
is easy to check that ${\cal C}$ is an additive self-dual code of type
$(2\kappa,\beta;\beta+\kappa-2\delta,\delta;\kappa)$ and $|{\cal C}|=2^{\kappa+\beta}$.
\end{demo}

Clearly, any of the statements $(i)-(vii)$ of Proposition
\ref{prop:antipodal} implies that $C=\Phi({\cal C})$ is antipodal. We
are going to see that the converse is not true.

\begin{lemm}\label{sd:minim4}
Let ${\cal C} \subset \mathbb{Z}_2^{\alpha}\times \mathbb{Z}_4^{\beta}$ ($\alpha,\beta \geq 1$) be an additive self-dual
code such that $C=\Phi({\cal C})$ is antipodal (i.e.
$(\u\mid{\mathbf{2}})\in{\cal C}$) and ${\cal C}_X$ is not self-dual, then
$\alpha\geq 4$ and $\beta \geq 4$.
\end{lemm}

\begin{demo}
Recall that $\alpha$ must be even for an additive self-dual code. We are
assuming that $(\u\mid{\mathbf{2}})\in{\cal C}$ and by Lemma \ref{sd:evenodd}
$({\mathbf{0}}\mid{\mathbf{2}})\in{\cal C}$, thus $(\u\mid{\mathbf{0}})\in{\cal C}$ implying
that any codeword has even weight in its binary coordinates.

If $\alpha=2$, then ${\cal C}_X=\{(0,0),(1,1)\}$, which is self-dual.
Therefore $\alpha\geq 4$.

Let $x=(x_b\mid x_q)$ and $y=(y_b \mid y_q)$ be two codewords such
that $x_b$ and $y_b$ are not orthogonal. Then $x_q$ and/or $y_q$
must have order 4; otherwise $x$ and $y$ would not be orthogonal.
Assume that $x_q$ has order 4. Since $w(x_b)$ is even, then $x_q$
has at least 4 coordinates of order 4, by Lemma \ref{sd:evenodd}.
Hence $\beta\geq 4$.
\end{demo}

If we assume that such an additive self-dual code ${\cal C}$ of type
$(4,4;\gamma,\delta;2)$ exists, the
sum of two codewords of order 4 always gives an order 2 codeword.
Hence it will have the same number of order 2 and order 4 codewords.
Thus
$$
2^{\gamma+\delta}=\frac{1}{2}\cdot 2^{\gamma+2\delta}
$$
and we obtain
that $\delta=1$. By Lemma \ref{sd:parameters}, we have
$\gamma + 2\delta = \beta + \kappa = 4+2=6,$ which
implies that $\gamma=4$.
Effectively such a code ${\cal C}$ exists. A generator matrix for
${\cal C}$ is
$$
\left (\begin{array}{cccc|cccc}
1 & 0 & 1 & 0 & 2 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 \\
0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 \\
\hline
0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{array}\right ).
$$
Therefore, we have proven the following result:

\begin{prop}
An additive self-dual code ${\cal C}$ with minimum cardinality and
number of coordinates such that $C=\Phi({\cal C})$ is antipodal and
${\cal C}_X$ is not self-dual is of type $(4,4;4,1;2)$.
\end{prop}

\bigskip
The following lemma give us a family of additive self-dual codes ${\cal C}$
such that $C=\Phi({\cal C})$ is antipodal and ${\cal C}_X$ is not
self-dual.

\begin{lemm} If $\delta < \kappa$, the $\Z_2\Z_4$-additive code ${\cal C}$ of type $(2\kappa,\beta;\beta+\kappa-2\delta,\delta;\kappa)$
with canonical generator matrix $${\footnotesize {\cal G}=\left (\begin{array}{cccccc|ccc}
    I_\delta  & {\mathbf{0}} & {\mathbf{0}} & I_\delta & {\mathbf{0}} & {\mathbf{0}} & 2I_\delta & {\mathbf{0}} & {\mathbf{0}} \\
        {\mathbf{0}} & 1     & {\mathbf{0}} & {\mathbf{0}}    & 1     & {\mathbf{0}} & {\bf 2} & {\mathbf{0}} & {\mathbf{0}} \\
       {\mathbf{0}}  & {\mathbf{0}} & I_{\kappa-\delta-1}& {\mathbf{0}} & {\mathbf{0}} & I_{\kappa-\delta-1} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\
       {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} &  {\mathbf{0}}  &  {\mathbf{0}} & {\bf 2} & 2I_{\beta-2\delta} & {\mathbf{0}} \\ \hline
       {\mathbf{0}}  & {\mathbf{0}} & {\mathbf{0}} & I_\delta & {\bf 1} & {\mathbf{0}} & I_\delta & \u & I_\delta \\
\end{array}\right )}$$ is an additive self-dual code if and only if $\beta -2\delta \equiv 2 \pmod 4$.
Moreover, $C=\Phi({\cal C})$ is antipodal and ${\cal C}_X$ is not self-dual.
\end{lemm}

\begin{demo}
Using the same arguments as in the proof of Lemma \ref{lemEx2}, we
have that ${\cal C}$ is an additive self-dual code if and only if
$\beta -2\delta \equiv 2 \pmod 4$. By Proposition
\ref{sd:propantipodal}, $C=\Phi({\cal C})$ is antipodal. And, it is
clear that ${\cal C}_X$ is not self-dual.
\end{demo}

redpajama_set_name: RedPajamaArXiv


On Thursday, nine Air France unions opposed the likely designation of Canadian Benjamin Smith at the helm of Air France-KLM; they find it "inconceivable" to appoint a foreign leader as CEO.
In a press release that was issued this morning, the unions – already conducting fifteen days of strike since February for better wages – state that it is inconceivable that Air France (French since 1933) falls into the hands of a foreign leader whose candidature has allegedly been pushed by a competing industrial group (read: Delta Air Lines).
Delta Air Lines, an American company, currently owns 8.8% of Air France-KLM. The French State is the largest shareholder with 14.3%.
"The successor of Jean-Marc Janaillac, the CEO that resigned in May after a majority of Air France staff voted "no" in the consultation on the pay agreement proposed on 16 April, must protect the interests of our national company and must have a good knowledge of the French social model", the group of unions announced, adding that the airline is in need of more transparency after "8 months of wandering without dialogue and without leadership".
The appointment of Ben Smith, Air Canada's COO and number two, is expected this afternoon just after the Air France-KLM board of directors meeting. "It seems that the fate of Air France must be sealed very discretely (in French: en catimini) during an extraordinary board meeting, and … via conference call," the unions added.


text:
Oslo Børs falls after vaccine news: – There is a lot of uncertainty now
Oil prices and stock markets around the world are turning downwards after new vaccine statements from the Moderna boss.
Erik Johansen / NTB
Updated less than 30 minutes ago
The main index on the Oslo Stock Exchange is down 1.2 percent after just under half an hour of trading, after a morning of falling oil prices and a weakening krone exchange rate.
At the stock exchange opening, the price of a barrel of North Sea oil (burnt) is down three percent for the day to 71.26 dollars a barrel.
This contributes to weighing on the oil companies Equinor and Aker BP, which are down 1.83 and 2.24 per cent, respectively. At the same time, heavyweights such as DNB, Hydro, Mowi and Telenor are in negative terrain.
Also on the stock exchanges in Asia, it turned to decline on Tuesday morning, while the arrows point downwards in the US pre-trade.
The European stock exchanges are also falling from the start, with all the Nordic stock exchanges in negative terrain. At the same time, the Paris and London stock exchanges are down around one percent in early trading.
The vaccine is probably not as effective
The global decline comes after Moderna boss Stéphan Blancet said the vaccine would probably be less effective against the omicron variant than the previous variants.
– The efficiency is not at the same level as we had with delta, he says Financial Times.
He also said that it could take months before vaccines against the new variant will be in production.
Vaccine chief warns of weaker effect against omicron
Think vaccine news pulls down
Senior strategist Joachim Bernhardsen in Nordea Markets says the markets will fall on Tuesday, mainly the statement from Moderna.
– They say the new coronavirus variant needs a new vaccine, and I think that is what pulls it down, Bernhardsen says to E24.
Bernhardsen points out that this means a longer period of measures, before a larger part of the population can be vaccinated.
– There is a lot of uncertainty now, and the markets do not like it. There is probably as much uncertainty behind it, says Bernhardsen and refers to today's market decline.
Senior strategist at Nordea Joachim Bernhardsen.
Number release from SAS
During the morning, SAS published its figures for the third quarter, and the shares listed on the Oslo Stock Exchange are down 1.9 percent from the start.
The airline delivered a profit of minus 744 million Swedish kroner in the accounting fourth quarter, compared to minus 2.57 billion at the same time last year.
At the same time, revenues amounted to SEK 5.76 billion in the quarter, from NOK 3.04 billion at the same time in 2020.
The Swedish krona is about the same as the Norwegian krone right now, so the amounts are roughly the same in Norwegian kroner.
Reference-www.aftenposten.no
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Source of historic victory: – Absolutely wild!
He could be one of the Olympics' biggest winners. But before the finish line, three major obstacles await.


Brimscombe and Thrupp will be without Karl Nash when they host Thatcham Town in the Hellenic Premier Division this Saturday, writes Johnny Morton.
Nash will serve a ban after being sent off in the Lilywhites' 4-0 win over Sun Sports in the extra preliminary round of the FA Cup last weekend.
Brimscombe will be hoping to go on another cup run after winning four games in the prestigious tournament last season, and manager Sam Prior was delighted with his sides dominant performance.
"We started really well and kept on the front foot and put them under so much pressure," said Prior.
"It was fantastic to score four and for Ollie to get four, but the all-round team performance was outstanding for the first 45 minutes, and in fairness, 4-0 flattered them in the end. We could have been five or six up at half-time we were that good.
"The sending off straight into the second half was disappointing and meant we had to play 40 minutes with 10 men, but the boys worked hard and it is credit to us that we created the better chances a man down. It was a dominant performance.
"We will play better sides without a doubt, but we can only beat what is in front of us and we did that comfortably."
Striker Ollie Pitt netted four first half goals, and after scoring 26 goals last season, Prior says Pitt will continue to be a key man for his side during this campaign.
"He's going to be very important this season," he added.
"His goal-scoring is not a problem for him. He proved last year when he got 26 that he can find the back of the net, and is a really key player for us. It is important for us that he continues to score goals and the others create chances for him.
"There's 10 other players who have to work hard for him to get those opportunities, and luckily for us on Saturday everything he hit seemed to go in the back of the net.
"The boys are working hard creating the chances for him, and it's not just about Ollie but the team as well, and if we all play well and he continues to score goals then I'll be happy, he will be happy and we will be going in the right direction."
Thatcham Town finished fourth in the league last season, and after recruiting new players during pre-season, Prior is expecting The Kingfishers to challenge for the title.
"I know Danny (Robinson, Thatcham manager) quite well, and we talk very regularly," he said.
"I know how strong they are and they've had a really good summer recruiting eight or nine new players. It's almost a new team for him, and they will certainly be up there come April.
"They are always a tough team to play. When they came to us last year they were 3-0 up in half an hour, so we have to make sure we start well or we will be punished again because that's what happens against the better sides.
"We are certainly in good enough shape to give a good account of ourselves and hopefully get something out of the game."


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"Nordic Farms Roof" oil on wood 12″ x 26″ . sold. Limited Edition (1/50) Giclee Prints Available.
"Story in the Clouds" oil on wood. 24″ x 36″ sold. Limited Edition (1/50) Giclee Prints Available.
"August From Mt. Philo" Oil on Wood. 9″ x 49″. Commission. Limited Edition (1/50) Giclee Prints Available.
"Last of the Golden Leaves" 12″ x 12″ oil on wood. sold. Limited Edition (1/50) Giclee Prints Available.
"Autumn Evening from Mt. Philo" 14″ x 40″ oil on wood. Commission. Limited Edition Giclee Prints Available.
"At Wilmarth Rd" 14″ x 22″ oil on wood. sold. Limited Edition Giclee Prints Available.
"Little Red Barns" oil on wood, 4″ x 12″ sold. Giclee Prints Available.
"Early Autumn From Mt. Philo" Oil on wood. 14″ x 40″. Sold. Limited Edition Giclee Prints Available.
"Mt. Mansfield in Manganese Blue Air" Oil on wood. 8″ x 12″. Sold. Prints Available.
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"Red Roof Meets Camel's Hump" oil on wood. 8″ x 10″ sold.
"Hay Collector by the Oak" oil on wood. 16″ x 20″ sold.
"Long Trail Forest" oil on wood. 20″ x 12″ sold.
"A Walk By the Dunes" oil on wood. 8″ x 14″ sold.
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"Rte 7 Sunset" oil on wood. 10″ x 40″ sold.
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I LOVE view from the bridge… Evening sky's like those are so magical. Glad you now have a place online to share your work, it's beautiful!
Your view from the bridge is the one piece of art we've taken the time to hang up in our new home. I look forward to seeing your art again at the Farmers market this summer.


text:
EntertainmentFeatured Stories
UIndy Gamers Club utilizes Discord to build community during COVID-19
by Noah Crenshaw | Editor-in-Chief March 10, 2021
written by Noah Crenshaw | Editor-in-Chief March 10, 2021
At the University of Indianapolis, there are more than 70 Registered Student Organizations that students can join to make new friends and further their own interests. One of these groups is the UIndy Gamers Club RSO.
Gamers Club is a place for people to come together and have fun, according to Gamers Club Public Relations Director and sophomore music major John Miley. Whether it is video games, board games or role-playing games, Gamers Club wants to make sure that people can come together, have fun and play games, Miley said.
Gamers Club Vice President and junior supply chain management major Samuel McGee said during their in-person meetings, the club tends to play more board games, but due to COVID-19, Gamers Club is doing online meetings and will play video games.
"We will do video game nights together," McGee said. "We have a Minecraft server. We do things like Codenames. We do things like the more complex board games, like Betrayal or things of that nature. We'll even play card games sometimes or any and all of the above. Whatever anyone wants to do, we are willing to do."
McGee said that due to moving to an online format, Gamers Club has had to become more creative with how they operate, meet and gain new members. Due to the change in gathering restrictions, Gamers Club is now able to do one in-person meeting a month, according to McGee. The rest of the month, the club meets virtually over Discord, an online instant messaging and digital distribution platform.
"Our online meetings are through Discord and we play games through that," McGee said. "We rely … more on our Discord server than anything else right now."
Miley said that the Gamers Club has been utilizing their Discord server to create online events for video games. The club will select three games a week and then see who joins the games, he said.
"We want to make sure not only are we advertising to people and not only are we trying to make sure everyone is playing games that they like, but we're also just trying to introduce new games [to] see how people feel about them and just have this be a full experience of playing new games while also going back and having fun with the old games that they really love," Miley said.
Photo Illustration by Jacob Walton
Although this is Miley's first year with Gamers Club, he said that based on what he has read about the club, they have not been able to do some of the activities they would usually do. One of these activities is an overnight event, and even if people wanted to do it, the club feels that mask mandates may be in jeopardy if they were to do so, Miley said. They also thought they would not be able to live up to the full extent of that event, he said.
Gamers Club meets for online events almost every Thursday from 9 p.m. to about midnight or 1 a.m., according to Miley and McGee. During these meetings, the club plays games and tries to use voice chat on Discord as they play, Miley said.
"The first week we did this, we decided we'll do Minecraft, Among Us and … Apex Legends," Miley said. "A lot of people play[ed] whichever game they wanted … Whenever we're not in-person, this has just become a weekly thing."
Miley said that because of COVID-19, it is hard for everyone on campus to be social. Gamers Club, however, has been an outlet that allows Miley to socialize and get people interested in having fun and playing games.
"For anyone who isn't a part of the club right now, I really would love them to join because I know the club would love to have more people," Miley said. "Gamers Club is, in my opinion, just a great social outlet that I personally have needed, and many other people in the club have needed during COVID."
Gamers Club is where McGee found his first group of friends when he came to UIndy, he said. McGee is originally from Buffalo, New York, and did not know anyone who went to UIndy.
"I was alone, but when I went to Gamers Club, [I found an] immediate group of friends, just all the comradery I could have asked for," McGee said. "My best friends today I've met at Gamers Club. I've learned a lot through Gamers Club and through the people I've met here, and it's sort of been like a community that's near and dear to my heart."
McGee said that Gamers Club has a great community and environment, whether it's in-person or over Discord. Everyone in the club is very friendly and there is not a lot of seriousness, he said.
"Everyone's very chill, just wanting to hang out, [and] play games," McGee said. "There's not a lot of seriousness, not a lot of officialness. It's a casual club for people who just want to hang out."
Gamers Club has been trying to plan events for the next few months, and will be announcing those plans in the future if the plans come to fruition, McGee said.Miley said that if anyone has any questions about joining Gamers Club, they can contact him at mileyj@uindy.edu. Students can also contact the club at gamersclub@uindy.edu.
Among UsApex LegendsBetrayalCodenamesCoronavirusCOVID-19COVID-19 pandemicDiscordIndianapolisIndyJohn MileyMinecraftNoah CrenshawSamuel McGeeThe ReflectorThe Reflector OnlineUIndyUIndy gamers clubUniversity of Indianapolis
UIndy Athletics to allow family spectators at outdoor events
Men's Soccer hits program record: No. 2 in the country
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Underground Theatre production 'Gruesome Playground Injuries' interlaces lives...
MADE IN THE A.M.
Senior Goodbye: Cassandra Lombardo


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