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\title{\textbf{On the number of nonequivalent propelinear extended perfect codes}}

\author{J. Borges\\
\small Department of Information and \\[-0.8ex]
\small Communica\-tions Engineering\\[-0.8ex]
\small Universitat Aut\`{o}noma de Barcelona\\[-0.8ex]
\small 08193-Bellaterra, Spain\\
\small \tt joaquim.borges@uab.cat
\and
I. Yu. Mogilnykh\\
\small Sobolev Institute of Mathematics and\\[-0.8ex]
\small Novosibirsk State University\\[-0.8ex]
\small  Novosibirsk, Russia.\\
\small\tt ivmog@math.nsc.ru
\and
J. Rif\`{a}\\
\small Department of Information and \\[-0.8ex]
\small Communica\-tions Engineering\\[-0.8ex]
\small Universitat Aut\`{o}noma de Barcelona\\[-0.8ex]
\small 08193-Bellaterra, Spain\\
\small \tt josep.rifa@uab.cat
\and
F. Solov'eva\\
\small Sobolev Institute of Mathematics and\\[-0.8ex]
\small Novosibirsk State University\\[-0.8ex]
\small  Novosibirsk, Russia.\\
\small\tt sol@math.nsc.ru
}

\date{\dateline{submission date}{acceptance date}\\
\small Mathematics Subject Classifications: 94B25, 94B60, 94A99
}




\begin{document}

\maketitle

\begin{abstract} The paper proves that there exists an exponential  number of nonequivalent propelinear extended perfect binary codes
of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov (2007) are
propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We
investigate the properties of these codes and show that any of them
has a normalized propelinear representation.

\bigskip\noindent \textbf{Keywords:} binary codes; extended perfect codes; normalized propelinear structures; propelinear codes
\end{abstract}

\section{Preliminaries}

Let $E_{q}$ be the set $\{0,1,\ldots,q-1\}$, where we distinguish
one element as $0$. We call \textit{words} the elements of the
cartesian product $E_q^n$. The word $(0,\ldots,0)$ is denoted by
$\zero$. Given two words $u=(u_1,u_2,\cdots, u_n),
v=(v_1,v_2,\cdots,v_n) \in E_q^n$, the \textit{Hamming distance}
$d(u,v)$ is the number of positions where they differ. In some
cases, when we are interested in an algebraic structure inside
$E_q$ we will take the $q$-ary finite field $\F_q$
%(or the Galois ring $GR(q)$)
instead of $E_q$, with $q=p^m$ and $p$ prime. The action
of an isometry of $E_{q}^n$ can be presented as the action of a
permutation $\pi$ on the coordinate positions $\{1,\ldots,n\}$ followed by the
action of $n$ permutations $\sigma_{1},\ldots,\sigma_{n}$ of
$E_{q}$:
%on coordinates $\{1,\ldots,n\}$:
$$\pi(x_{1},\ldots,x_{n})=(x_{\pi^{-1}(1)},\ldots,x_{\pi^{-1}(n)}),$$
$$(\sigma_{1},\ldots,\sigma_{n})(x_{1},\ldots,x_{n})=(\sigma_{1}(x_{1}),\ldots,\sigma_{n}(x_{n})).$$

%%Coordinate wise
The permutation $\sigma = (\sigma_{1},\ldots,\sigma_{n})$ is called
a {\it multi-permutation}. The composition $\sigma\circ
\sigma'$ of two multi-permutations $\sigma$ and $\sigma'$ is
the following multi-permutation: $(\sigma_{1}\circ
\sigma'_{1},\ldots,\sigma_{n}\circ \sigma'_{n})$, where $\sigma_{i}\circ
\sigma'_{i}$ is the composition $\sigma_{i}\circ
\sigma'_{i}(x_i)=\sigma_{i}(
\sigma'_{i}(x_i))$, for any $i\in\{1,2,\ldots,n\}$.

 By $(\sigma;\pi)(x)$ we denote the image of $x$ under
an isometry $(\sigma;\pi)$ :

$$(\sigma;\pi)(x)=\sigma(\pi(x)).$$

%((\sigma_{1}(x_{\pi(1)}),\ldots,\sigma_{n}(x_{\pi(n)})).$$

A $q$-ary {\it code} $C$ of length $n$ is a subset of $E_{q}^n$. We
denote by $\iso(C)$ the largest subgroup of isometries
%%
of $E_{q}^n$ that fix
the code $C$ and we call it the {\it isometry group} of the
code $C$.

\begin{defi}\label{q-ary_propelinear}
A $q$-ary code $C$ of length $n$, $\zero \in C$, is called {\em
propelinear} if for any codeword $x$ there exists a coordinate
permutation $\pi_x$ and %%
a multi-permutation $\sigma_x = (\sigma_{x,1},\ldots,\sigma_{x,n})$ satisfying:
\begin{itemize}
\item[(i)] for any $x\in C$ it holds  $(\sigma_{x};\pi_{x})(C)=C$ and  $(\sigma_{x};\pi_{x})(\zero)=x$,
%%    (the property of a transitivity);
\item[(ii)] if $y\in C$ and $z=(\sigma_{x};\pi_{x})(y)$, then:\\ $\pi_{z}=\pi_{x}\circ \pi_{y}$ and $\sigma_{z,i}=\sigma_{x,
i}\circ \sigma_{y,\pi^{-1}_{x}(i)},$ for any $i \in
\{1,\ldots,n\}$; or, equivalently,
$(\sigma_{z};\pi_{z})=(\sigma_{x};\pi_{x})(\sigma_{y};\pi_{y}).$
%$\sigma_z=\sigma_x\pi_x\sigma_y\pi_x^{-1}$.
\end{itemize}
\end{defi}

A $q$-ary code is called {\em transitive} if the  isometry group of the code acts transitively on its codewords, i.\,e., the code satisfies the property $(i)$ in Definition \ref{q-ary_propelinear}. Transitive codes are studied in \cite{S2004,S2005}.

In the binary case, when $q=2$, taking the usual addition on
$E_2=\F_2$, the above definition is reduced to the following:

A binary code $C$ is {\em propelinear} \cite{Rif1} if for each
$x\in C$ there exists a coordinate permutation $\pi_x$ such that:
\begin{itemize}
\item[(i)] $x+\pi_x(C)=C$;
\item[(ii)] if $x+\pi_x(y)=z$, then $\pi_z=\pi_x\circ \pi_y$, for any
$y\in C$.
\end{itemize}

%We can define the following group operation on codewords of a
%propelinear code: $$x\star
%y=(\sigma_x;\pi_x)(y)=(\sigma_{x,1}(y_{\pi_{x}(1)}),\ldots,\sigma_{x,n}(y_{\pi_{x}(n)})).$$

As in the binary case, where we can define a group structure $\star$ on $C$
which is compatible with the Hamming distance, that is, such that
$d(x\star u,x\star v)=d(u,v)$, also in the $q$-ary case, given a $q$-ary propelinear code we define the operation $\star$ as
\begin{equation}\label{star}
x \star v = (\sigma_x;\pi_x)(v)\;\;\;\mbox{for any } x\in C,\;\;\mbox{for any }
v\in E_q^n.\end{equation}

%The next Lemma give us some properties of $q$-ary propelinear codes.
In \cite{BR,Rif3,Rif1,Rif2}, properties of binary propelinear
codes are deeply studied. Linear codes and $\Z_2\Z_4$-linear codes are propelinear but, perhaps, one much more interesting example of propelinear code is the original Preparata code~\cite{Rif2} wich is not a $\Z_4$-linear code (although there is a $\Z_4$-linear code with the same parameters~\cite{kh}). In \cite{BRS2011,BMRS2011}, the
relations between classes of propelinear and transitive codes are
investigated. The problem of distinguishing these classes had been
open since 2006. In \cite{BMRS2011} it was established that these classes are different. In fact, it was found that the binary Best code of length 10 is transitive, but not
propelinear.

In this paper we establish a new lower bound
$\frac{1}{8n^2\sqrt{3}}e^{\pi\sqrt{2n/3}}(1+o(1))$ on the number
of nonequivalent propelinear extended perfect binary codes of
length $4n$ for $n$ going to infinity. This bound is obtained by
showing propelinearity of transitive Potapov codes~\cite{Pot}, the
rank of which is one more than the rank of the extended Hamming
code of the same length. The previous lower bound on the number of
nonequivalent propelinear extended perfect binary codes of length
$n=2^m, m\geq 4$  was $\lfloor \log_2
(n/2)\rfloor ^2,$ see \cite{BRS2011,BMRS2011}. Despite
the fact that the new class of propelinear codes is larger than
the old class from \cite{BRS2011,BMRS2011}, it does not cover the
old one (the ranks of codes from \cite{BRS2011,BMRS2011} and from \cite{Pot} do not coincide), so the result \cite{BRS2011,BMRS2011} keeps current. We
investigate in this paper the properties of new propelinear codes
and show that any of them has a normalized propelinear
representation.


Now, we give a generalization of the most relevant
properties of propelinear codes to the $q$-ary case.

Let $C$ be a propelinear code; let $\Pi$ and $\Sigma$ be the set
of permutations and the set of multi-permutations, respectively, assigned to
the codewords of $C$ (Definition~\ref{q-ary_propelinear}) and let
$\star$ be the afore defined operation in $C$. To emphasize this particular propelinear structure of $C$ we write $(C,\Pi,\Sigma,\star)$ or, simply $(C,\star)$ when we do not require any information
about the set of associated permutations.

The next lemmas are easy to prove from elementary group theory.

\begin{lemm}\label{previs}
Let $(C,\Pi,\Sigma,\star)$ be a $q$-ary propelinear code of length $n$.
\begin{itemize}
\item[(i)] Let $x\in C$ and $u,v\in E_q^n$. If $x\star u=x\star v$, then $u=v$.
\item[(ii)] The all-zeroes word $\zero$ is a codeword, $\zero\in C$.
\item[(iii)] For any codeword $x\in C$, there exists a unique codeword $x'\in C$ such that $x\star x'=\zero$.
\end{itemize}
\end{lemm}

%\begin{proof}
%For (i) note that $x\star u=x\star v$ is equivalent to
%$\sigma_x(\pi_x(u))=\sigma_x(\pi_x(v))$ and hence $u=v$.
%For (ii), since $x\star C=C$ (by Definition
%\ref{q-ary_propelinear}), we have that it must be a codeword $e\in
%C$ such that $x\star e=x$. We also know that $x\star \zero=x$ (by
%Definition \ref{q-ary_propelinear}). Therefore, by (i), we have
%that $e=\zero$ and $\zero\in C$. Finally, condition (iii) comes easily from $x\star C=C$ and (ii).
%\end{proof}

\textbf{Note:} Not always the defined operation $\star$ can be
generalized in a proper way over all $E_q^n$. That is, from $x, y
\in C$, $u\in E_q^n$ such that $x\star u = y \star u$, we do not
necessarily have $x=y$.

\begin{lemm}
Let $(C,\Pi,\Sigma,\star)$ be a $q$-ary propelinear code. Then $C$ equipped with this operation $\star$ is a
group.
\end{lemm}

%\begin{proof}
%Clearly, the operation $\star $ is closed on $C$. Associativity follows from associativity of the group formed by
%isometries with respect to composition. Indeed, let $x,y,z$ be any three codewords. We have:
%
%$x \star (y\star z)$=$(\sigma_x;\pi_x)((\sigma_y;\pi_y)(z))$=
%$(\sigma_x;\pi_x)(\sigma_y;\pi_y)z=$
%$((\sigma_x;\pi_x)(\sigma_y;\pi_y))z=$ $(x\star y)\star z.$
%
%
%%\begin{eqnarray*}
%%(x\star y)\star z &=& \left(\sigma_x(\pi_x(y))\right)\star z=(\sigma_x\pi_x\sigma_y\pi_x^{-1}\pi_x\pi_y)(z)\\
%%        &=& (\sigma_x;\pi_x)(\sigma_y(\pi_y(z)))=x\star (y\star z).
%%\end{eqnarray*}
%Given the
%equality $x\star y=z,$ where $x,y,z\in C$, we know that $x$ and $y$
%uniquely determine $z$. Also, by Lemma \ref{previs}, we know that
%$x$ and $z$ uniquely determine $y$. Assume that $w\in C$ and
%$w\star y=z$. According to (iii) of Lemma \ref{previs}, let $y'\in
%C$ be the codeword such that $y\star y'=\zero$. Thus, we have the following:
%$$
%x\star y=w\star y ,$$
%$$x\star y\star y'=w\star y\star
%y',$$
%$$x\star \zero=w\star \zero,$$
%and therefore, $x=w.$
%
%Thus, $C$ equipped with the operation $\star$ is a group.
%\end{proof}

 Note that, apart from the group structure on $C$ given by the operation $\star$, there can exist
a lot of different group structures on a propelinear code, including
nonisomorphic ones (for binary case see \cite{BMRS2011}).

Clearly, $\zero$ is the identity element in $(C,\star )$ and we
denote by $x^{-1}$ the inverse element of the codeword $x$. Denote
by $\mbox{Id}_n$ the identity permutation over any set of cardinal
$n$. Now, we can link the coordinate permutations and the
multi-permutations of inverse elements.

\begin{lemm}
Let $(C,\star)$ be a $q$-ary propelinear code. Then,
\begin{itemize}
\item[(i)] The codeword $\zero$ has the identities as the associated coordinate permutation $\pi_\zero=\mathrm{Id}_n$ and
multi-permutation $\sigma_\zero=(\mathrm{Id}_q,
\mathrm{Id}_q,\ldots, \mathrm{Id}_q)$, respectively.
\item[(ii)]
We have $\pi_{x^{-1}}=\pi_x^{-1}$ and
$\sigma_{x^{-1},i}=\sigma_{x,\pi_x(i)}^{-1}$, for any codeword $x\in C$ and any $i \in \{1,\ldots,n\}$.
%$\sigma_{x^{-1}}=\pi_x^{-1}\sigma_x\pi_x$.
\end{itemize}
\end{lemm}

%\begin{proof}
% Since $x\star \zero=x$ for all $x\in C$, we have
% the equality
% $(\sigma_{x},\pi_{x})=(\sigma_{x},\pi_{x})(\sigma_{0},\pi_{0})$,
% that implies:
%$\pi_\zero=\mbox{Id}_n$, $\sigma_\zero=(\mathrm{Id}_q, \mathrm{Id}_q,\ldots, \mathrm{Id}_q)$.
%
% For (ii), from the equality $x\star x^{-1}=\zero$ we obtain that the
% isometry
% $(\sigma_{x^{-1}},\pi_{x^{-1}})$ is the inverse of
% $(\sigma_{x},\pi_{x})$. Hence, from Definition~\ref{q-ary_propelinear}, $\pi_{x^{-1}} = {\pi_x}^{-1}$ and, for any $i\in \{1,\ldots,n\}$, $\mathrm{Id}_q = \sigma_{x,i} \circ \sigma_{x^{-1},\pi_{x^{-1}}^{-1}(i)} = \sigma_{x,i} \circ \sigma_{x^{-1},\pi_{x}(i)}$.
%
%% For the values of $\sigma_{x^{-1}}$ and $\pi_{x^{-1}}$
%%defined in the statement and for any word $y$ from $E_{q}^{n}$
%%we have:
%%
%%$$(\sigma_{x};\pi_{x})(\sigma_{x^{-1}};\pi_{x^{-1}})(y)=
%%(\sigma_{x};\pi_{x})(\sigma_{x,\pi_{x}(1)}^{-1},\ldots,\sigma_{x,\pi_{x}(n)}^{-1})(y_{\pi_{x}(1)},\ldots,y_{\pi_{x}(n)})=$$
%%$(\sigma_{x};\pi_{x})(\sigma_{x,\pi_{x}(1)}^{-1}(y_{\pi_{x}(1)}),\ldots,\sigma_{x,\pi_{x}(n)}^{-1})(y_{\pi_{x}(n)})=$
%%$\sigma_{x}\sigma_{x}^{-1}(y)=y.$
%\end{proof}

Finally, as we said before, the action of $\star $ over $E_q^n$ is
Hamming distance compatible.

\begin{lemm}
Let $(C,\star )$ be a $q$-ary propelinear code. Then,
$$
d(x\star u,x\star v)=d(u,v)\;\;\;\mbox{for any } x\in C,\;\;\mbox{for any } u,v\in
E_q^n.
$$
\end{lemm}

%\begin{proof}
%Straightforward since coordinate permutations and multi-permutations are distance preserving maps.
%\end{proof}


\section{Isotopically propelinear MDS codes}
A $q$-ary code of length $n$, satisfying the property $(i)$ in
Definition~\ref{q-ary_propelinear} with $\pi_x=\mbox{Id}_n$ for
any $x$ in the code is called an {\it isotopically transitive
code}. A notion of isotopic transitivity was introduced by Potapov
in \cite{Pot} and used for constructing an exponential number of
nonequivalent transitive extended binary perfect codes of length
$n$ as $n$ goes to infinity. We call a $q$-ary propelinear
structure on a code $C$ of length $n$ {\em isotopically
propelinear}, if for any $x \in C$ it holds $\pi_x=\mbox{Id}_n$.
If there is an isotopically propelinear structure on a code $C$,
we call $C$ {\it isotopically propelinear}.

A $q$-ary code $C$ of length $m$ with minimum distance 2
of size $q^{m-1}$ is a kind of MDS code. All MDS-codes
we use in this paper are of this kind. A {\it quaternary
MDS code} is a code with $q=4$. A function
$f:E_{q}^{m-1}\rightarrow E_{q}$ is called a $(m-1)$-ary
{\it quasigroup} of order $q$ if $f(x_{1},\ldots,x_{m-1})\neq
f(y_{1},\ldots,y_{m-1})$ for any words $(x_{1},\ldots,x_{m-1})$
and $(y_{1},\ldots,y_{m-1})$ from $E_{q}^{m-1}$ that differs in
only one position.

%The following construction shows that there exist isotopically
%propelinear MDS codes.
%\begin{prop}
%Let $<E_{q},\cdot>$ be a group, $M$ be the MDS code that
%corresponds to quasigroup
%$f(a_{1},\ldots,a_{m})=a_{1}\cdot\ldots\cdot a_{m}$. Then $M$ is
%isotopically propelinear code.
%\end{prop}
%\begin{proof}
%Let $a$ be a codeword of $M$. Define the multi-permutation
%$\sigma_{a}:$

%$\sigma_{a,1}(\alpha)=\alpha\cdot a_{1}$

%\end{proof}


It is known that there exists a one-to-one correspondence between
$(m-1)$-ary quasigroups of order $q$ and MDS $q$-ary codes of
length $m$. Given a $(m-1)$-ary quasigroup $f$ we can construct the
code $\{(x,f(x)):x \in E_{q}^{m-1}\}$.

%In the rest of this section we  use $E_4=\{0,1,2,3\}$ as a set of four elements.

Concerning $MDS$ codes, we mainly consider the quaternary case, i.e.,
$q=4$. We use two different group operations in $E_4$. First, we
use $*$ to refer to the addition when we see the elements in $E_4$
as elements in $\Z_4$. Second, we use $\oplus$ to refer to the
addition when we see the elements in $E_4$ as elements in
$\Z_2\times \Z_2$ through the Gray map given by $0\rightarrow
(0,0), 1 \rightarrow (0,1), 2\rightarrow (1,1), 3\rightarrow
(1,0)$.

%Note that the operations $\star$ and $*$ are different.
Both next examples were used in~\cite{Pot} to construct extended perfect transitive codes.
\smallskip

{\bf Example 1.} Let us consider the function $x_1 \ast x_2$ from
$E_{4}^{2}$ to $E_{4}$. From the correspondence between MDS codes
and quasigroups we have that $\{(x_1,x_2,x_1 \ast x_2): x_1,x_2\in
E_{4}\}$ is a MDS code. It is straightforward to see that this
code is an isotopically propelinear code with the corresponding
permutations $\sigma_{x,1}(y)=x_{1}*y$, $\sigma_{x,2}(y)=x_{2}*y$,
$\sigma_{x,3}(y)=x_{3}*y$ for any $y\in E_4$, where $x_3=x_1 \ast
x_2$.
\medskip

{\bf Example 2.} Let $x_{1}\oplus x_{2}$ be the function from
$E_{4}^{2}$ to $E_{4}$. The corresponding MDS code is isotopically
propelinear with  the following permutations
$\sigma_{x,1}(y)=x_{1}\oplus y$, $\sigma_{x,2}(y)=x_{2}\oplus y$,
$\sigma_{x,3}(y)=x_{3}\oplus y$ for $y\in E_4$, where $x_3=x_1
\oplus x_2$.



Potapov~\cite{Pot} proved the isotopic transitivity of quaternary MDS
codes obtained from an isotopically transitive MDS code $M$ and
the MDS code from Example 2 using the following concatenated
construction:
\begin{equation}\label{MDScomp}
\begin{split}
\{(x_{1},\ldots,x_{i-1},y_{1},\ldots,y_{r},&x_{i+1},\ldots,x_{m}):\\
&y_{1}\oplus y_{2}\oplus \ldots\oplus y_{r}=x_{i}, \,\,
x=(x_{1},\ldots,x_m)\in M\},
\end{split}
\end{equation}
for some fixed $i$,
$1\leq i\leq m-1$, and for any $r=1,2,\ldots$

If the initial code corresponds to a
quasigroup $f$, that is,
$M=\{(x,f(x)):x \in E_{4}^{m-1}\}$ then the constructed code
corresponds to the following composition of the quasigroup $f$ and
the quasigroup from Example 2:
\begin{equation*}
\begin{split}
g(x_{1},\ldots,x_{i-1},&y_{1},\ldots,y_{r},x_{i+1},\ldots,x_{m-1})=\\
&f(x_{1},\ldots,x_{i-1},y_{1}\oplus
\ldots\oplus y_{r},x_{i+1},\ldots,x_{m-1}).
\end{split}
\end{equation*}

The main result of this section is Proposition~\ref{L1}, where we
show that the constructed code is, in fact, isotopically
propelinear given that $M$ is isotopically propelinear. Below,
without restricting generality, $i$ is equal to 1.

First of all we recall two technical lemmas.

\begin{lemm}\cite{Pot}\label{lem_pot} Let $\varphi$ be a permutation on the
elements of $E_{4}$.
% such that $\varphi(0)=0$.
Then
$\varphi(a\oplus b)=\varphi(a)\oplus \varphi(b) \oplus \varphi(0).$
\end{lemm}

Given a permutation $\sigma$ on the elements of
$E_{4}$ and a word $y=(y_{1},\ldots,y_{r})$ in $E_{4}^{r}$ such
that $y_{1}\oplus \ldots \oplus y_{r}=\sigma(0)$ we define the
permutations $\tau_{y,1},\ldots,\tau_{y,r}$ in
$E_{4}$ in the following way:
\begin{equation}\label{tau}
\tau_{y,s}(\alpha)=\sigma(\alpha)\oplus y_{1} \oplus \ldots \oplus
y_{r} \oplus y_{s} = \sigma(\alpha)\oplus \sigma(0) \oplus y_{s}, \,\mbox{where $s\in \{1,2,\ldots,r\}.$}
\end{equation}

The above defined permutations satisfy the following statement:
%%
\begin{lemm}\cite[Prop. 7]{Pot}\label{Proptau}
%\begin{enumerate}
 %   \item $(\tau_{y,
%1}(0),\ldots,\tau_{y,
%r}(0),\sigma(0))=(y_{1},\ldots,y_{r},y_{1}\oplus\ldots\oplus
%y_{r}).$
%\item
For any $x_{1},\ldots,x_{r}\in E_{4}$ we have $\tau_{y,1}(x_{1})\oplus\ldots\oplus \tau_{y,r}(x_{r})=\sigma(x_{1}\oplus\ldots\oplus x_{r}).$
%\end{enumerate}\underbrace{}
\end{lemm}
%\begin{proof}
%From Lemma~\ref{lem_pot}, $\tau_{y,1}(x_{1})\oplus\ldots\oplus \tau_{y,r}(x_{r}) =
%\sigma(x_1) \oplus \ldots \oplus \sigma(x_r) \oplus \underbrace{\sigma(0)\oplus \ldots \oplus\sigma(0)}_r
%\oplus \\y_1\oplus \ldots \oplus y_r = \sigma(x_{1}\oplus\ldots\oplus x_{r}) \oplus \sigma(0)\oplus \sigma(0) =
%\sigma(x_{1}\oplus\ldots\oplus x_{r}).$
%\end{proof}

%In particular, the previous proposition implies isotopic
%transitivity of MDS code
%$$\{y_{1},\ldots,y_{m},y_{1}\oplus\ldots\oplus y_{m}:y_{i}\in
%\F_{4}, i=1\ldots m\}.$$


\begin{prop}\label{L1} Let $M$ be a quaternary $\mathrm{MDS}$ code of length
$m$ with  an  isotopically propelinear structure $(M,\Sigma,\star)$  and $M'=\{(y_{1},\ldots,y_{r},x_{2},\ldots,x_{m}):(y_{1},\ldots,y_{r})\in
E_{4}^{r}, \,\, y_{1}\oplus y_{2}\oplus \ldots\oplus y_{r}=x_{1},
\,\, (x_{1},\ldots,x_{m})\in M\}.$

Then $(M',\Delta,\star)$ is an
isotopically propelinear structure on the $\mathrm{MDS}$ code $M'$
with the mul\-ti\--per\-mutation
$\delta_{z}\!=\!(\tau_{y,1},\ldots,\tau_{y,r},\sigma_{x,2},\ldots,\sigma_{x,m})$ assigned to
$z\!=\!(y_{1},\ldots,y_{r},x_{2},\ldots,x_{m})$, where $\tau_{y,s}$ is
defined in (\ref{tau}) taken $\sigma_{x,1}$ as the permutation
$\sigma$, for any $s\in \{1,2,\ldots,r\}$.
\end{prop}
\begin{proof}
It is easy to see that the code $M'$ has minimum distance 2,
length $m+r-1$ and size $4^{r+m-2}$, i.e., it is an MDS code over
$E_{4}$. Recall that, by  definition, $\sigma_{x,1}$ is the
multi-permutation assigned to a codeword $x=(x_1,\ldots,x_m).$


By definition of a propelinear structure if a codeword $z$ of $M'$ is obtained from a
codeword $x$ of $M$ by replacing
the first coordinate $x_{1}$ with the sequence of elements $y_{1},\ldots,y_{r}$ from $E_4$,
such that $y_{1}\oplus\ldots\oplus y_{r}=x_{1}$, then
$\delta_{z}=(\tau_{y,1},\ldots,\tau_{y,r},\sigma_{x,2},\ldots,\sigma_{x,m})$.

The code $M'$ equipped with the permutations defined above was
proved to be isotopically transitive, see \cite{Pot}. Now, we show
that
 this structure is isotopically propelinear. In order to do so, following Definition~\ref{q-ary_propelinear}, we need to show that $\delta_{\delta_{z}(z')}=\delta_{z}\circ\delta_{z'}$, where $\delta_z, \delta_{z'},\delta_{\delta_z(z')}$ are the assigned permutations to the elements $z,z', \delta_z(z')$, respectively.

 Let $z=(y_{1},\ldots ,y_{r},x_{2},\ldots,x_{m})$ and $z'=(y'_{1},\ldots ,y'_{r},x'_{2},\ldots,x'_{m})$ be two codewords of
 $M'$, i.e.
$$ y_{1}\oplus\ldots\oplus y_{r}=x_{1},$$
 \begin{equation}\label{y}
 y'_{1}\oplus\ldots\oplus y'_{r}=x'_{1},
 \end{equation}
where  $x=(x_{1},\ldots,x_{m}), x'=(x'_{1},\ldots,x'_{m})\in M$.
The permutations assigned to $z$ and $z'$ are:
$$\delta_{z}=(\tau_{y,1},\ldots,\tau_{y,r},\sigma_{x,2},\ldots,\sigma_{x,m})$$
$$\delta_{z'}=(\tau_{y',1},\ldots,\tau_{y',r},\sigma_{x',2},\ldots,\sigma_{x',m}).$$




%Permutations
%$$(\tau_{y,1},\ldots,\tau_{y,r},\sigma_{x,2},\ldots,\sigma_{x,m})
%\,\, \mbox{and} \,\,
%(\tau_{y',1},\ldots,\tau_{y',r},\sigma_{x',2},\ldots,\sigma_{x',m})$$
%such that
%$$\tau_{y,s}(\alpha)=\sigma_{x,1}(\alpha)\oplus y_{1} \oplus
%\ldots\oplus y_{r}\oplus y_{s}, $$
%\begin{equation}\label{tauteo}
%\tau_{y',s}(\alpha)=\sigma_{x',1}(\alpha)\oplus y'_{1} \oplus
%\ldots\oplus y'_{m}\oplus y'_{s},
%\end{equation}
%$ s=1,\ldots,r$ respectively, are assigned to codewords $z$ and $z'$ respectively.
%  We  show that
% $\delta_{\delta_{z}(z')}=\delta_{z}\circ\delta_{z'}$.

 We have
$\delta_{z}(z')=(\tau_{y,1}(y_{1}'),\ldots,\tau_{y,r}(y_{r}'),\sigma_{x,2}(x_{2}'),\ldots,\sigma_{x,m}(x_{m}'))$ and, using Lemma~\ref{Proptau}:
$\tau_{y,1}(y'_{1})\oplus\ldots\oplus\tau_{y,r}(y'_{r})=\sigma_{x,1}(y'_{1}\oplus
\ldots\oplus y'_{r})$. From this equality and (\ref{y}) we see
that $\delta_{z}(z')$ is obtained by substituting the first
coordinate of
$\sigma_{x}(x')=(\sigma_{x,1}(x'_{1}),\ldots,\sigma_{x,m}(x'_{m}))$
with the sequence of elements $\tau_{y,1}(y'_{1}),\ldots,\tau_{y,r}(y'_{r})$ and
$\tau_{y,1}(y'_{1})\oplus\ldots\oplus\tau_{y,r}(y'_{r})=\sigma_{x,1}(x'_{1})$. Therefore, $\delta_{z}(z')$ belongs to $M'$.
 From the isotopic propelinearity of $(M,\Sigma,\star)$, we have that the permutation $\sigma_{x}\circ \sigma_{x'}$ is assigned to the codeword
 $\sigma_{x}(x')$ of $M$, so the multi-permutation
 %, by definition of permutations on $M'$ we have that
$\delta_{\delta_{z}(z')}$ coincides with $\delta_{z}\circ\delta_{z'}$ in each one of the $j$th positions, for $r+1\leq j \leq m+r-1$.

For the first $r$ positions, by the definition of $(M',\Delta,\star)$, we have
\begin{equation}\label{tau3}
\delta_{\delta_{z}(z'),s}(\alpha)=\tau_{\delta_{z}(z'),s}(\alpha)=\sigma_{x,1}\circ\sigma_{x',1}(\alpha)\oplus\tau_{y,1}(y'_{1})
\oplus\ldots\oplus\tau_{y,r}(y'_{r})\oplus\tau_{y,s}(y'_{s})
\end{equation} for $s=1,\ldots,r.$
It remains to prove that the permutation ($\ref{tau3}$) coincides with $\tau_{z,s}\circ\tau_{z',s}$ for
$s=1,\ldots,r$.
For any $s$ above,  using Lemma~\ref{lem_pot} and Lemma~\ref{Proptau}, the above equality (\ref{tau3}) comes to:
\begin{equation*}
\begin{split}
\tau_{\delta_{z}(z'),s}(\alpha) &=\sigma_{x,1}(\sigma_{x',1}(\alpha))\oplus \sigma_{x,1}(y_{1}'\oplus\ldots\oplus y'_r)\oplus \sigma_{x,1}(y'_s)\oplus\sigma_{x,1}(0)\oplus y_{s}\\
&= \sigma_{x,1}(\sigma_{x',1}(\alpha))\oplus \sigma_{x,1}(x_{1}'\oplus y'_s)\oplus y_{s}\\
&= \sigma_{x,1} \big(\sigma_{x',1}(\alpha)\oplus x'_1 \oplus y'_s \big) \oplus \sigma_{x,1}(0)\oplus y_{s}\\
&= \tau_{z,s}(\sigma_{x',1}(\alpha)\oplus \sigma_{x',1}(0) \oplus y'_s)= \tau_{z,s}(\tau_{z',s}(\alpha)).\\
\end{split}
\end{equation*}
\end{proof}

Potapov \cite{Pot} considered quasigroups of the following form:
$$f(x_{1},\ldots,x_{n-1})=(x_{1}\oplus\ldots\oplus
x_{i_{1}})\ast(x_{i_{1}+1}\oplus\ldots\oplus x_{i_{2}})\ast\ldots
\ast(x_{i_{m-2}+1}\oplus\ldots\oplus x_{n-1}),$$ where $1\leq
i_{1}\leq \ldots \leq i_{m-1}\leq n-1$ (in throughout what
follows, we denote this quasigroup with
$f_{i_{1},\ldots,i_{m-2}}$), and proved the transitivity property
of any MDS code corresponding to a quasigroup of this type. In
this section we show the isotopic propelinearity of these MDS
codes.

Indeed, let $M$ be the code
$\{(y_{1},\ldots,y_{m-1},y_{1}\ast\ldots\ast y_{m-1}):y_{j}\in
E_{4}, j=1,2,\ldots, m-1\}$. This code is isotopically propelinear
with the permutation
$\sigma_{y}=(\sigma_{y,1},\ldots,\sigma_{y,m-1},\sigma_{y,m})$
assigned to the codeword
$y=(y_{1},\ldots,y_{m-1},y_{1}\ast\ldots\ast y_{m-1})$, where
$\sigma_{y,j}(\alpha)=\alpha\ast y_{j}$, for $1\leq j\leq m-1$ and
$\sigma_{y,m}(\alpha)=\alpha \ast y_{1}\ast \ldots\ast y_{m-1}$.
In order to obtain the code $M'=\{(x,f_{i_{1},\ldots,i_{m-2}}(x):
x \in E_{4}^{n-1} \}$ we apply $m-1$ times the construction
(\ref{MDScomp}) to every coordinate $j$, $1\leq j\leq m-1$. By
Proposition~\ref{L1}, the code $M'$
%,
%corresponding to the quasigroup
%$f(x_{1},\ldots,x_{n-1})=(x_{1}\oplus\ldots\oplus
%x_{i_{1}})\ast(x_{i_{1}+1}\oplus\ldots\oplus x_{i_{2}})\ast\ldots
%\ast(x_{i_{m-2}+1}\oplus\ldots\oplus x_{n-1})$
is isotopically propelinear. In other words, we obtain the
following statement.

%The explicit expressions for the multi-permutations assigned to the codewords in $M'$.

\begin{coro} \label{sigmaMDS}
Let $M'=\{(x,f_{i_{1},\ldots,i_{m-2}}(x)): x \in E_{4}^{n-1}\}$.
Then there exists an isotopically propelinear structure
$(M',\Sigma,\star)$, with
 the
multi-permutation $\sigma_{x}$ assigned to a codeword $x$ being such that
\begin{equation}
\sigma_{x,i_{j}+t}(\alpha)=(\alpha\ast
(x_{i_{j}+1}\oplus\ldots\oplus x_{i_{j+1}}))\oplus x_{i_{j}+t},
\end{equation}
for $1 \leq t \leq i_{j+1}-i_{j}$ and $0\leq j \leq m-2$, $i_{0}=0$.
\end{coro}

As a consequence of the isotopic propelinearity of these codes we have
the same lower bound for the number of nonequivalent isotopically
propelinear codes as the one in \cite{Pot} for the isotopically
transitive codes.


\begin{coro}\label{number isotopically propelinear MDS}
There  exist at least
$\frac{1}{4(n-1)\sqrt{3}}e^{\pi\sqrt{2(n-1)/3}}(1+o(1))$
nonequivalent quaternary isotopically propelinear $\mathrm{MDS}$
codes of length $n$, for $n$ going to infinity.
\end{coro}

%%
\section{Propelinear extended perfect codes}

In this section we prove that binary extended perfect Phelps codes
\cite{P84} constructed from isotopically propelinear MDS codes are
propelinear.

First of all, we give some additional notations from \cite{Pot} and prove some
necessary statements. Let $C_{0}$ be the binary extended  Hamming
code of length 4:

$$C_{0}=\{(0,0,0,0),(1,1,1,1)\}.$$

Bellow, we follow the same notations as in~\cite{Pot}. We
identify the elements $0,1,2,3$ of $E_4$ with  the $4$th, $1$st, $2$nd, $3$rd coordinate positions of
$\F_2^4$, respectively, so we have
  $e_0=(0,0,0,1)$,
$e_1=(1,0,0,0)$ $e_2=(0,1,0,0)$ $e_3=(0,0,1,0)$.
Now, define the codes in $\F_2^4$:
\begin{equation}\label{car}
C_{a}^{r}=C_{0}+(1+r)e_{0}+e_{a}, \,\, \mbox{for } r\in\{0,1\}, \,\, a \in
E_{4}.
\end{equation}
%%

The codes $\{C_{a}^{r}\}_{r=0,1;a \in E_{4}}$ give a partition of
$\F_{2}^{4}$ into extended perfect codes and the codes
$\{C_{a}^{0}\}_{a \in E_{4}}$ give a partition of the binary full
even weight code into extended perfect codes.

All  extended perfect codes of length 4 can be represented as the cosets of $C_{0}$ and, more specifically, if $b$ is in $C_{a'}^{r'}$ then:
\begin{equation}\label{sum}
b+C_{a}^{r}=C_{a\oplus a'}^{r+r'}.
\end{equation}

Let $S_4$ be the symmetric group of permutations over $E_4$.
%, and extend the action of $S_4$ over $E_4^n$ by defining $\pi(x_1,x_2,\ldots, x_n) = (x_{\pi(1)}, x_{\pi(2)}, \ldots, x_{\pi(n)})$.
In \cite{Pot} it was shown that the action of a  permutation of
coordinates (fixing the 4th coordinate) on the partition
$\{C_{a}^{0}\}_{a \in E_{4}}$ can be represented as the action of
a permutation of the cosets in this partition.
%of coordinates $\sigma \in S_4$ corresponds to a permutation of cosets.

\begin{prop}\label{sigmapi}
For every $\sigma \in S_4$ the following holds:

 $$C_{\sigma(a)}^{r}+e_{\sigma(0)}+e_{0}=\pi(C_{a}^{r}),$$
\mbox{for all $a \in E_{4}$ and $r\in\{0,1\}$}, where $\pi(\alpha)=\sigma(\alpha)\oplus \sigma(0)$ fixes the $4$th coordinate.

\end{prop}

\begin{proof}
Using (\ref{car}), we get that
$$C_{\sigma(a)}^{r}+e_{\sigma(0)}+e_{0}=C_0+(1+r)e_0+e_{\sigma(a)}+e_{\sigma(0)}+e_0.$$

Since $e_0+e_{\sigma(a)}+e_{\sigma(0)}+e_{\sigma(0)\oplus\sigma(a)}$
is the all-zeros or the all-ones vector, we obtain the desired property:

$$C_{\sigma(a)}^{r}+e_{\sigma(0)}+e_{0}=C_0+(1+r)e_0+e_{\pi(a)}=\pi(C_0+(1+r)e_0+e_{a})=\pi(C_a^r).$$

\end{proof}

%%


\begin{prop}\label{propsigmapihom}

The mapping $\sigma\rightarrow \pi$ is a homomorphism from $S_4$ to $S_3$. The kernel of the homomorphism is the set of permutations of type $\sigma(\alpha)=\alpha\oplus b,$ for some $b \in E_4$ and for any $\alpha$.
\end{prop}

\begin{proof}~ Let $\pi(\alpha)=\sigma(\alpha)\oplus\sigma(0)$ and $\pi'(\alpha)=\sigma'(\alpha)\oplus\sigma'(0)$ for any $\alpha \in E_4$.
Using Lemma \ref{lem_pot},  we have

$$\pi'(\pi(\alpha))=\sigma'(\sigma(\alpha)\oplus\sigma(0))\oplus \sigma'(0)=\sigma'(\sigma(\alpha))\oplus\sigma'(\sigma(0))$$
for any $\alpha \in E_4$.

The claim about the kernel is trivial.

\end{proof}
%%
Now, consider  the Phelps concatenation construction \cite{P84}, see also \cite{Z76}:

\begin{equation}\label{Phelps}
C=\bigcup_{(h_{1},\ldots,h_{n})\in
H}\bigcup_{(a_{1},\ldots,a_{n})\in M}
C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}},
\end{equation}
where $H$ is an extended Hamming code of length $n$, $M$ is a
quaternary MDS code of length $n$ and codes $C_{a_{i}}^{h_{i}},
i=1,\ldots,n,$ are defined in (\ref{car}). Using the construction
(\ref{Phelps}), Potapov in \cite{Pot} found a large class of
transitive codes taking $M$ being isotopically transitive. %%
These  MDS codes  correspond to quasigroups
\begin{equation}  f(x_{1},\ldots,x_{n-1})=(x_{1}\oplus\ldots\oplus
x_{i_{1}})\ast(x_{i_{1}+1}\oplus\ldots\oplus x_{i_{2}})\ast\ldots
\ast(x_{i_{m-2}+1}\oplus\ldots\oplus x_{n-1}),\end{equation} for
any $i_{1},\ldots,i_{m-2}$, such that $1\leq i_{1}<\ldots <
i_{m-2} < n-1$. In the previous section we proved that all these
isotopically transitive %%
MDS codes are isotopically propelinear. Now, we show that all
Potapov's transitive extended perfect binary codes are propelinear
too.

\begin{theo}\label{Phelps.theo} Let $M$ be a quaternary isotopically propelinear MDS code of length
$n$, $H$ be a binary extended Hamming code of length $n$. Then,
the code
$$ C=\bigcup_{(h_{1},\ldots,h_{n})\in
H}\bigcup_{(a_{1},\ldots,a_{n})\in M}
C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}}
$$
is a binary propelinear extended perfect code of length $4n$.
\end{theo}
\begin{proof} Let $(M,\Sigma,\star)$ be an isotopically propelinear
structure on the code $M$. Let
$\sigma_{a}=(\sigma_{a,1},\ldots,\sigma_{a,n})$ be the
multi-permutation assigned to a codeword $a=(a_{1},\ldots,a_{n})$ of $M$. For any $i\in
\{1,\ldots,n\}$, let $\pi_{a_{i}}$ be the permutation defined by
Proposition \ref{sigmapi} when $\sigma$ is equal to
$\sigma_{a,i}$:

$$C_{\sigma_{a,i}(b)}^{r}+e_{a_{i}}+e_{0}=\pi_{a_{i}}(C_{b}^{r}), \,\, \mbox{for any b} \in E_{4}, \,\, \mbox{and any} \,\, r\in\{0,1\}.$$

To every codeword $c$ in the class
$C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}},$ where
$(h_{1},\ldots,h_{n})\in H$ we assign the permutation
$\pi_{a}=(\pi_{a_{1}},\ldots,\pi_{a_{n}})$ acting on $4n$
coordinates in the following way: if $x=(x_{1},\ldots,x_{n})$ is a
word of length $4n$ such that $x_{i}$ is a word of length 4
for any $i$, then
$\pi_{a}(x_{1},\ldots,x_{n})=(\pi_{a_{1}}(x_{1}),\ldots,\pi_{a_{n}}(x_{n}))$.
In \cite{Pot} it is proved that the code $C$ with these
permutations is transitive. We now show that it is propelinear too.


Let $c \in C_{a_{1}}^{h_{1}}\times\ldots\times
C_{a_{n}}^{h_{n}}$;  $c'\in C_{a'_{1}}^{h'_{1}}\times\ldots\times C_{a'_{n}}^{h'_{n}}$ and let  $\pi_{a}$ and $\pi_{a'}$ be the permutations
assigned to the codewords $c$ and $c'$, respectively. To show that $C$ is propelinear it is enough to
show that  the permutation assigned to the class
$c+\pi_{a}(C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n}}^{h'_{n}})$ is $\pi_{a}\circ \pi_{a'}$.

Let us find  more convenient representation for the class
$c+\pi_{a}(C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n}}^{h'_{n}})$. By the definition of $\pi_{a}$ we have the
following equalities:

$$\pi_{a}(C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n}}^{h'_{n}})=\pi_{a_{1}}(C_{a'_{1}}^{h'_{1}})\times\ldots\times
\pi_{a_{n}}(C_{a'_{n}}^{h'_{n}})=$$
$$(C_{\sigma_{a,1}(a'_{1})}^{h'_{1}}+e_{a_{1}}+e_{0})\times\ldots\times
(C_{\sigma_{a,n}(a'_{n})}^{h'_{n}}+e_{a_{n}}+e_{0}).$$
Since $c \in C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}}$,
 from the last equality and using (\ref{sum}) we obtain

$$c+\pi_{a}(C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n}}^{h'_{n}})=C_{\sigma_{a,1}(a'_{1})}^{h'_{1}+h_{1}}\times\ldots\times
C_{\sigma_{a,n}(a'_{n})}^{h'_{n}+h_{n}}.$$


Thus, we have  to find the permutation corresponding to the
codewords of the class
$C_{\sigma_{a,1}(a'_{1})}^{h'_{1}+h_{1}}\times\ldots\times
C_{\sigma_{a,n}(a'_{n})}^{h'_{n}+h_{n}}$. By propelinearity of
$M$, the multi-permutation assigned to
$(\sigma_{a,1}(a'_{1}),\ldots,\sigma_{a,n}(a'_{n}))\in M$
is
$\sigma_{a}\circ\sigma_{a'}=(\sigma_{a,1}\circ\sigma_{a',1},\ldots,
\sigma_{a,n}\circ\sigma_{a',n})$. Finally, from Proposition
\ref{propsigmapihom} and the definitions of permutations on $C$, we
obtain that $\pi_{a}\circ\pi_{a'}$ is the permutation assigned to the
codewords of the class
$C_{\sigma_{a,1}(a'_{1})}^{h'_{1}+h_{1}}\times\ldots\times
C_{\sigma_{a,n}(a'_{n})}^{h'_{n}+h_{n}}=c+\pi_{a}(C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n}}^{h'_{n}})$.
 \end{proof}

And, finally, considering MDS codes corresponding to quasigroups of type
$$f_{i_{1}\ldots i_{m-2}}(a_{1}\ldots a_{n-1})\!\!=\!\!(a_{1}\oplus\ldots\oplus
a_{i_{1}})\ast(a_{i_{1}+1}\oplus\ldots\oplus a_{i_{2}})\ast\ldots
\ast(a_{i_{m-2}+1}\oplus\ldots\oplus a_{n-1}),$$ applying the
results of the previous section, and using the same considerations as
in \cite{Pot} we obtain:

\begin{coro} There exist  at least
$\frac{1}{8n^2\sqrt{3}}e^{\pi\sqrt{2n/3}}(1+o(1))$  nonequivalent propelinear extended perfect binary codes of length $4n$, for $n$ going to infinity.
\end{coro}

\section{Normality}
The concept of binary normalized propelinear codes was introduced
in \cite{BMRS2011}.
 A propelinear structure on a binary code $C$
is called {\it normalized} if the codewords of the same coset of
the code $C$ by the kernel have the same assigned permutation.

In this section we analyze the propelinear structure defined in
Theorem \ref{Phelps.theo}, when MDS code corresponds to the
quasigroup $$f_{i_{1}\ldots i_{m-2}}(a_{1}\ldots a_{n-1})\!\!=\!\!(a_{1}\oplus\ldots\oplus
a_{i_{1}})\ast(a_{i_{1}+1}\oplus\ldots\oplus a_{i_{2}})\ast\ldots
\ast(a_{i_{m-2}+1}\oplus\ldots\oplus a_{n-1}).$$ We show that the structure is normalized if and only if $m$ is
odd. For even $m$ it is not normalized, however we can find an exponential number of propelinear representations of
the Phelps codes, which are normalized propelinear.

%First of all, we describe the kernel of Phelps code
%\begin{equation} C=\bigcup_{(h_{1},\ldots,h_{n})\in
%H}\bigcup_{(a_{1},\ldots,a_{n})\in M}
%C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}}.
%\end{equation} Consider a
%codeword $c'$ of a class $C_{a'_{1}}^{h'_{1}}\times\ldots\times
%C_{a'_{n}}^{h'_{n}}$ and a class
%$C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}}$
% of the code $C$. From (\ref{sum}), we see that

% $$c+C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}}=
%C_{a_{1}\oplus a'_{1}}^{h_{1}+h'^{1}+1}\times\ldots\times
%C_{a_{n}\oplus a'_{n}}^{h_{n}+h'^{n}+1}.$$

By $\mathrm{Ker}(M)$, the {\it kernel} of an arbitrary MDS code
$M$ over $E_{4}$ we mean the collection of all its codewords $a$
such that
$$a\oplus M=M.$$
We begin  with describing the kernel of the MDS and Phelps
codes.

\begin{prop}\label{propMDSPh}
Let $M$ be a $\mathrm{MDS}$ code of length $n$, $H$ be an extended
Hamming code of length $n$,
$$C=\bigcup_{h\in H}\bigcup_{a\in M}
C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n}}^{h_{n}}.$$ Then a
codeword $c$  from the code $C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n}}^{h'_{n}}$ belongs to $\mathrm{Ker}(C)$ if and only if
the word $a'=(a'_{1},\ldots,a'_{n})$ belongs to $\mathrm{Ker}(M)$.
\end{prop}
\begin{proof}
Let $c$ be a codeword in $C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n}}^{h'_{n}}$. We know that $(h'_1,h'_2,\ldots, h'_n) +H =H$, so
$$
C=\bigcup_{h\in H}\bigcup_{a\in M}
C_{a_{1}}^{h_{1}+h'_1}\times\ldots\times C_{a_{n}}^{h_{n}+h'_n}.
$$

From (\ref{sum}) we have:

$$c+C=c+\bigcup_{h\in
H}\bigcup_{a\in M} C_{a_{1}}^{h_{1}+h'_1}\times\ldots\times
C_{a_{n}}^{h_{n}+h'_n}=\bigcup_{h\in H}\bigcup_{a\in
M}C_{a_{1}\oplus a'_{1}}^{h_{1}}\times\ldots\times C_{a_{n}\oplus
a'_{n}}^{h_{n}}.$$ Therefore, $c$ is in $\mathrm{Ker}(C)$ if and
only if $a'$ is in $\mathrm{Ker}(M)$.
\end{proof}
%\begin{prop}
% A codeword of a class
%$C_{a'_{1}}^{h'_{1}}\times\ldots\times C_{a'_{n}}^{h'_{n}}$
% belongs to the kernel of code $C$ if and only if
% $a'=(a'_{1},\ldots,a'_{n})$ is such that:

% $$a'\oplus M=M,$$
% where $M$ is the MDS code used to construct Phelps code and
%$a\oplus b=(a_{1}\oplus b_{1},\ldots,a_{n}\oplus b_{n})$.
%\end{prop}

From this point and further on we represent a codeword of a MDS
code corresponding to a quasigroup $f$ as $(a,f(a))$.

\begin{prop}\label{kerMDS}
Let $f$ be a (n-1)-quasigroup of order 4 and $M$ the
$\mathrm{MDS}$ code $M=\{(a,f(a)): a \in E_{4}^{n-1}\}$. The
codeword $(a',f(a'))$ belongs to $\mathrm{Ker}(M)$ if and only if
$f(a\oplus a')=f(a)\oplus f(a')$, for all $a \in E_{4}^{n-1}$.

\end{prop}
\begin{proof}
If $(a',f(a'))\in \mathrm{Ker}(M)$ then $(a',f(a')) \oplus M=M$,
so, for all $a\in E_4^{n-1}$ we have $(a'\oplus a, f(a')\oplus
f(a)) \in M$. So $(a'\oplus a, f(a')\oplus f(a))$ and $(a\oplus
a', f(a\oplus a'))$ both belong to $M$, therefore, $f(a)\oplus
f(a')=f(a\oplus a')$ and vice versa.
\end{proof}


Now, we focus on the case when MDS code corresponds to the
quasigroup
$$f_{i_{1}\ldots i_{m-2}}(a_{1}\ldots a_{n-1})\!\!=\!\!(a_{1}\oplus\ldots\oplus
a_{i_{1}})\ast(a_{i_{1}+1}\oplus\ldots\oplus a_{i_{2}})\ast\ldots
\ast(a_{i_{m-2}+1}\oplus\ldots\oplus a_{n-1}).$$

We need a technical lemma. In through out what follows,
$u^{-1}$ denotes the inverse element of $u$ in the group
$(E_{4},\ast)$.

\begin{lemm}~\label{func}

\begin{itemize}
\item[(i)] For all $u\in E_4$ it is true $u'\oplus u = \left\{ \begin{array}{ll}
u' \ast u & \mbox{for } u'\in \{0,2\}; \\
u'\ast u^{-1} & \mbox{for } u'\in \{1,3\}.
\end{array}\right.$
\item[(ii)] There is no $u'\in E_{4}$ such that
the equality
$$u'\oplus (u\ast v)=u'\ast u^{-1}\ast v$$ holds for any $u$ and $v$ from $E_{4}$.
\end{itemize}
\end{lemm}

\begin{proof}
The first statement follows directly from the definitions of operations $\oplus$ and
$\ast$.

Let us prove the second statement. If $u'$ belongs to $\{0,2\}$, then by the first statement we have  $u' \oplus (u\ast
v)=u'\ast u\ast v$ for any $u$ and $v$ in $E_{4}$, but for $u \in
\{1,3\}$ we have: $u'\ast u\ast v\neq u'\ast u^{-1}\ast v$.

If $u'$ belongs to $\{1,3\}$, then by the first statement we obtain  $u' \oplus
(u\ast v)=u'\ast u^{-1}\ast v^{-1}$ for any $u$ and $v$ in
$E_{4}$, but for $v \in \{1,3\}$ we have $u'\ast u^{-1}\ast
v^{-1}\neq u'\ast u^{-1}\ast v$.
\end{proof}

\medskip

We now describe the kernel of a particular MDS code.
\begin{prop}\label{propp}
Let $M=\{(a,a_{1}\ast\ldots\ast a_{m-1}):a=(a_1,\ldots,a_{m-1})
\in E_{4}^{m-1}\}$.
 Then
 $$\mathrm{Ker}(M)\!=\!\left\{%
\begin{array}{l}
    \{(a',a'_{1}\ast\ldots\ast a'_{m-1})\mid\!  a'\in \{0,2\}^{m-1}\}, \mbox{ if m is odd,} \\
\{(a',a'_{1}\ast\ldots\ast a'_{m-1})\mid\!  a'\in \{0,2\}^{m-1}\cup \{1,3\}^{m-1}\}, \mbox{ if m is even.}
    \end{array}%
\right.     $$
\end{prop}
\begin{proof}
By Proposition \ref{kerMDS} it is true that $(a',a'_{1}\ldots
a'_{m-1})\in \mathrm{Ker}(M)$ if and only if for any $a \in E_{4}^{m-1}$:
\begin{equation} \label{kerp1}
(a'_{1}\oplus a_{1})\ast\ldots \ast(a'_{m-1}\oplus
a_{m-1})=(a'_{1}\ast\ldots\ast a'_{m-1})\oplus
(a_{1}\ast\ldots\ast a_{m-1}).
\end{equation}
Let the first $k$ coordinates of $a'$ be from $\{0,2\}$ and the
last $m-1-k$ coordinates be from $\{1,3\}$. Now, from Lemma \ref{func}
we can express the operation $\oplus$ in (\ref{kerp1}) by the
operation $\ast$: $(a'_{1}\oplus a_{1})\ast\ldots
\ast(a'_{m-1}\oplus a_{m-1})=a'_{1}\ast a_{1}\ast\ldots \ast
a'_{k}\ast a_{k}\ast a'_{k+1}\ast a^{-1}_{k+1}\ast \ldots\ast
a'_{m-1}\ast a^{-1}_{m-1},$ so the condition  (\ref{kerp1}) of
belonging to the kernel of $M$ can be rewritten as:
$$(a'_{1}\ast\ldots\ast a'_{m-1})\oplus (a_{1}\ast\ldots\ast a_{m-1})=
a'_{1}\ast\ldots\ast a'_{m-1}\ast a_{1}\ast \ldots \ast a_{k}\ast a^{-1}_{k+1}\ast \ldots \ast a^{-1}_{m-1},$$
for any $a \in E_{4}^{m-1}$.

    Let $u'$ be equal to $a'_{1}\ast\ldots\ast a'_{m-1}$; $u$ be equal to $a_{1}\ast\ldots\ast
    a_{k}$ and  $v$ be equal to $a_{k+1}\ast\ldots\ast a_{m-1}$. Using
    these notations, the property of belonging to the kernel is
    equivalent to:
\begin{itemize}
\item[(i)] If $k=0$, then $a'\in \{1,3\}^{m-1}$ and $u'\oplus v= u' * v^{-1}$ implying, by Lemma \ref{func}, that $u'\in \{1,3\}$. Therefore $m$ is even.
\item[(ii)] If $k=m-1$, then $a'\in \{0,2\}^{m-1}$ and $u'\oplus u = u'*u$ implying, by Lemma \ref{func}, that $u'\in \{0,2\}$, which is true for any $m$.
\item[(iii)] If $0<k<m-1$, then $u'\oplus (u*v)=u'*u*v^{-1}$, which is impossible, again according to Lemma \ref{func}.
\end{itemize}

%$$u'\oplus v=u'\ast v, \mbox{for } v\in E_{4} \mbox{ if } k=1,$$
%$$u'\oplus v= u'\ast v^{-1}, \mbox{for } v\in E_{4} \mbox{ if }
%k=m+1,$$
%$$u'\oplus (u\ast v)=u'\ast v^{-1}\ast u, \mbox{for } u, v\in E_{4} \mbox{ if } k\neq1,m+1.$$
%
%From Lemma \ref{func} and the definition of the operation $\ast$ the
%first option is done for any $a'\in \{0,2\}^{m+1}$, the second
%one is done for any $a'\in \{1,3\}^{m+1}$ only for even $m$
%and the third one is never realized.
\end{proof}
%In throughout what follows $a$ is a word of length $n-1$ from $E_{4}^{n-1}.$

For the general case we have the following description for the kernels:
\begin{theo}\label{T2}
Let $M=\{(a,f_{i_{1},\ldots,i_{m-2}}(a)): a \in E_{4}^{n-1}\}$ be
a $\mathrm{MDS}$ code. \\Then, $(a,f_{i_{1},\ldots,i_{m-2}}(a))$
belongs to $\mathrm{Ker} (M)$ if and only if  the word of partial
sums
$$(\oplus_{j=1}^{i_{1}}a_{j},\oplus_{j=i_{1}+1}^{i_{2}}a_{j},\ldots,\oplus_{j=i_{m-2}+1}^{n-1}a_{j})$$
belongs to $\{0,2\}^{m-1}$ for odd $m$ and to $\{0,2\}^{m-1}\cup
\{1,3\}^{m-1}$ for even $m$.
\end{theo}
\begin{proof}
Directly from Proposition \ref{propp}.
\end{proof}
From Theorem \ref{T2} and Proposition \ref{propMDSPh} we
obtain:

\begin{coro}\label{kersize}
Let $C$ be the code obtained by  Phelps construction
$$C=\bigcup_{h\in H}\bigcup_{(a,f_{i_{1},\ldots,i_{m-2}}(a))\in M}
C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n-1}}^{h_{n-1}}\times
C_{f_{i_{1},\ldots,i_{m-2}}(a_{1},\ldots,a_{n-1})}^{h_{n}}.$$

 If $m$ is odd
then $|\mathrm{Ker}(C)|=2^{3n-2-\log_{2}(n)}$ and if $m$ is even
$|\mathrm{Ker}(C)|=2^{3n-1-\log_{2}(n)}$.
\end{coro}

\begin{theo}\label{PhelpsNormal}
Let $$C=\bigcup_{h\in H}\bigcup_{(a,f_{i_{1},\ldots,i_{m-2}}(a))\in M}
C_{a_{1}}^{h_{1}}\times\ldots\times C_{a_{n-1}}^{h_{n-1}}\times
C_{f_{i_{1},\ldots,i_{m-2}}(a_{1},\ldots,a_{n-1})}^{h_{n}},$$ and let $(C,\Pi,\star)$ be the propelinear structure on $C$, defined
in Theorem \ref{Phelps.theo}. Then

(i) $(C,\Pi,\star)$ is normalized, if $m$ is odd;


(ii) $(C,\Pi,\star)$ is not normalized and there exist at least $2^{n-2}$ different normalized propelinear structures on $C$, if $m$ is even.

%and the
%kernel $(K,\star)$ has a subgroup of index $2$, isomorphic to
%$\Z_{2}^\ell$, for some natural $\ell$.

\end{theo}

\begin{proof}
Let $M'$ be the MDS code $\{(a,f_{i_{1},\ldots,i_{m-2}}(a)): a\in E_{4}^{n-1}\}$ and $(a',f_{i_{1},\ldots,i_{m-2}}(a'))\in
\mathrm{Ker}(M')$.
Let
the multi-permutation $(\sigma_{a',1},\ldots,\sigma_{a',n-1},\sigma_{a',n})$
be assigned to the word $(a',f_{i_{1},\ldots,i_{m-2}}(a'))$ of $M'$. Let $t$ be such that
$i_{s}+1\leq t \leq i_{s+1}$, for $s=0,\ldots, m$, $i_{0}=0,
i_{m-1}=n-1$. Then, by the definition of the propelinear structure
on $M'$, see Corollary \ref{sigmaMDS}, we have
$$\sigma_{a',t}(\alpha)=((a_{i_{s}+1}\oplus\ldots\oplus a_{i_{s+1}})\ast \alpha)\oplus
a_{t}\oplus a_{i_{s}+1}\oplus\ldots\oplus a_{i_{s+1}},$$
$$\sigma_{a',n}(\alpha)=(a_{1}\oplus\ldots\oplus
a_{i_{1}})\ast(a_{i_{1}+1}\oplus\ldots\oplus a_{i_{2}})\ast\ldots
\ast(a_{i_{m-2}+1}\oplus\ldots\oplus a_{n-1})\ast \alpha.$$

By Theorem \ref{T2}, $a_{i_{s}+1}\oplus\ldots\oplus a_{i_{s+1}}$
belongs to $\{0,2\}$ or $\{1,3\}$ simultaneously for any
$s=0,\ldots,m-2$.

Let $a_{i_{s}+1}\oplus\ldots\oplus a_{i_{s+1}}$ be from
$\{0,2\}$. Then, by Lemma \ref{func},
$\sigma_{a',t}(\alpha)=(a_{i_{s}+1}\oplus\ldots\oplus
a_{i_{s+1}})\oplus \alpha \oplus a_{t}\oplus
a_{i_{s}+1}\oplus\ldots\oplus a_{i_{s+1}}=\alpha\oplus a_{t}$.
 Therefore,
according to the assignment of permutations to the codewords of
$C$ (see Theorem \ref{Phelps.theo}) and by Proposition \ref{propsigmapihom} (ii),  the
permutation
$\pi_{(a',f_{i_{1},\ldots,i_{m-2}}(a'))}=(\pi_{(a',f_{i_{1},\ldots,i_{m-2}}(a')),1},\ldots,\pi_{(a',f_{i_{1},\ldots,i_{m-2}}(a')),n})$
assigned to a codeword $c$ in the class
$C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n-1}}^{h'_{n-1}}\times C_{f_{i_{1},\ldots,i_{m-2}}(a')}^{h'_{n}}$ is the identity. Taking into account the description of kernel given in Theorem \ref{T2},
the structure is normalized for odd $m$.


Let $a_{i_{s}+1}\oplus\ldots\oplus a_{i_{s+1}}$ be from the set
$\{1,3\}$. Then, by Lemma \ref{func},
$\sigma_{a',t}(\alpha)=((a_{i_{s}+1}\oplus\ldots\oplus
a_{i_{s+1}})\oplus \alpha^{-1} \oplus a_{t}\oplus
a_{i_{s}+1}\oplus\ldots\oplus a_{i_{s+1}}=\alpha^{-1}\oplus
a_{t}$. It is easy to see that the permutation $\sigma_{a',t}(\alpha)=\alpha^{-1}\oplus a_{t}$
 is not a permutation of ``linear type", i.e. cannot be expressed as
$\alpha\oplus u$ for some fixed $u$ from $E_{4}$. So, according to
the assignment of permutations to the codewords of $C$ (see
Theorem \ref{Phelps.theo}), and again by Proposition \ref{propsigmapihom}(ii), the
permutation
$\pi_{(a',f_{i_{1},\ldots,i_{m-2}}(a'))}=(\pi_{(a',f_{i_{1},\ldots,i_{m-2}}(a')),1},\ldots,\pi_{a',f_{i_{1},\ldots,i_{m-2}}(a'),n})$
assigned to the codeword $c$ in
$C_{a'_{1}}^{h'_{1}}\times\ldots\times
C_{a'_{n-1}}^{h'_{n-1}}\times C_{f_{i_{1},\ldots,i_{m-2}}(a')}^{h'_{n}}$ is not the
identity.


Hence, for even $m$, one half of the codewords of the kernel have
assigned the identity permutation, and the second half have assigned the same non-identity permutation.

Note that in case of even $m$, there exist several normalized propelinear structures.
According to the proof of the Theorem~\ref{Phelps.theo},
every codeword is assigned a permutation of the form
 $\pi_{a}=(\pi_{a_{1}},\ldots,\pi_{a_{n}})$, where $\pi_{a_i}\in S_4$
fixes one coordinate (see Proposition \ref{sigmapi}), for any
$i=1,\ldots,n$. Hence, $\pi_{a_i}$ could only have order 2 or 3,
for all $i=1,\ldots,n$, so $\pi_a$ has order 2, 3 or 6. However,
orders 3 and 6 can not occur. Given a propelinear code
$(D,\Pi,\star)$ with $|D|$ being a power of 2, we have that
$|\Pi=\{\pi_{x}:x \in C\}|$ is also a power of 2. Indeed, the map
$x \mapsto \pi_x$ is group homomorphism from the group $D$ onto
the permutation group $\Pi$. Therefore, for any codeword $x$, the
order of $\pi_x$ is a power of 2 as well. We conclude that all
permutations assigned to the codewords of $C$ are involutions and,
in this case, $C$ has $|\{\pi_{x}:x \in
\mathrm{Ker}(C)\}|^{\log(|C|)-\mathrm{dim}(\mathrm{Ker}(C))}$
different normalized propelinear structures (a proof of this fact
is given in \cite{BMRS2011}). Substituting the value of
$\mathrm{dim}(\mathrm{Ker}(C))$ obtained from Corollary
\ref{kersize}, we obtain $2^{n-2}$ different structures.
\end{proof}

 \section*{Acknowledgments}
 This work was supported in part by the Spanish
MICINN under Grants  MTM\-2009-08435 and TIN2010-17358, and by the Catalan AGAUR under Grant 2009SGR\-1224.
The second author was supported by  the Grants RFBR 12-01-00448,  12-01-31098 and
13-01-00463. The work of the fourth author was partially
supported by the Grant RFBR 12-01-00631-a.

The authors are grateful to the referee for the useful remarks which allowed to improve the presentation of the results.

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