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\begin{document}

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\titleofpresentation{Permutation invariance of binary codes and linearity of Reed-Muller codes over some rings}

\presenter{Bahattin YILDIZ}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

By starting with an open conjecture from [2] in which the authors
conjectured that the Reed-Muller codes $RM(r,m)$ won't be
$\mathbb{Z}_4$-linear for non-trivial cases, we try to answer the
same question for the ring $\mathbb{F}_2+u\mathbb{F}_2$. Linear
codes over this ring have been studied in works like [3], [4] and
[5]. In this work we will be proving that the Reed-Muller codes
$RM(r,m)$ are $\F_2+u\F_2$-linear contrary to the case of $\Z_4$.
In fact we will generalize this further to some other rings.

The tools that we will be using are going to be relating
$\F_2+u\F_2$-linearity as well as linearity over these other rings
that we will consider of binary codes with permutation invariance
where we consider some suitable permutation group in each case.
These will help us understand these linear codes better.

A linear code over a ring $R$ of length $n$ is just an
$R$-submodule of $R^n$ and a permutation acts on a linear code by
switching the coordinates. In this aspect linear codes are
algebraic objects and while proving our results we will just use
these aspects of them.

The main rings to be used are going to be $\F_2+ u\F_2$ which is
constructed via $u^2=0$, $\F_2+ u\F_2+u^2\F_2+ u^3\F_2$
 which is constructed via $u^4 = 0$ and its generalizations. We
 will also be considering the ring $\F_2+ u\F_2+v\F_2+ uv\F_2$
 which is constructed via $u^2 = v^2 = 0, uv= vu.$ We will
 demonstrate that being linear over the ring $\F_2+ u\F_2+u^2\F_2+\dots +
 u^{2^k-1}\F_2$of a binary code of length $2^kn$
is equivalent to being invariant under the permutation group
$\Z_{2^k}$ and we will also show that linearity over the ring
$\F_2+ u\F_2+v\F_2+ uv\F_2$ of a binary code of length $4n$ means
being invariant under the permutation group $K_4$, where $K_4$
denotes the Klein-4 group. After these results we will finally
consider the case of Reed-Muller codes $RM(r,m)$ and we will
answer the question posed by the authors in [2] for the rings that
we have considered above.



\begin{ourbiblio}{99}  %  Delete this environment if you
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\bibitem{yourname1}B.~Yildiz, \emph{Weight Enumerators and Gray maps of Linear codes over
rings}, \textbf{Thesis}, California Institute of Technology,
(2006)

\bibitem{Sloane} A.R. ~Hammons, V.~Kumar, A.R.~Calderbank, N.J.A.~Sloane, and P.
~Sol\'{e}, \emph{The $\\mathbb{Z}_4$-linearity of Kerdock,
Preparata, Goethals and related codes}, IEEE Trans. Inform.
Theory,

\bibitem{Gaborit1} S.T.~Dougherty, P.~Gaborit, M.~Harada, A.~Munemasa and P.~Sol\'{e},
 \emph{Type IV self-dual codes over rings}, IEEE Trans.
 Inform. Theory, \textbf{45} (1999), 2345--2360.

 \bibitem{Gaborit2} S.T.~Dougherty, P.~Gaborit, M.~Harada and P.~Sol\'{e}, \emph{Type
II codes over $\mathbb{F}_2+u\mathbb{F}_2$}, IEEE Trans. Infrom.
Theory, \textbf{45} (1999), 32--45.

\bibitem{Betsumiya} K. Betsumiya, S.Ling, F.R. Nemenzo, \textit{Type II Codes over $\F_{2^m}+u\F_{2^m}$}, Discrete Math.
textbf{275} (2004), 43--65.

\end{ourbiblio}

\presenteraddress{Department of mathematics, Fatih University}  % The name of a university is fine.

\email{byildiz@fatih.edu.tr}

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