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{{\bf Dual codes of translation planes
}}

J. D. Key\\ Department of Mathematical Sciences\\
Clemson University\\ Clemson SC 29634
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{\bf Abstract}
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Early in the study of codes associated with finite projective planes
it was shown that the $p$-ary code of a projective plane of order $n$,
where $p$ is a prime dividing $n$, has minimum weight $n+1$ and the
codewords of minimum weight are the scalar multiples of the incidence
vectors of the lines.

For the dual code, neither the minimum weight nor the nature of the
possible minimum words, is known in the general case, even though
these are the codes that are most useful in applications since they
can be decoded using majority logic decoding. Various bounds can be
established, and for some particular classes precise results are
known. In particular, for desarguesian planes of even order $q=2^m$,
where $p=2$, the minimum weight is $q+2$ and the minimum words are the
incidence vectors of the hyperovals, which always exist in the
desarguesian planes.

When $p$ is an odd prime, even for the desarguesian planes the minimum
weight of the dual is not in general known, except for the case when
the order is prime, in which case the minimum weight is $2p$.

In this talk we will discuss the existing known bounds in the case $p$
odd, and show how these can be improved for translation planes
(including the desarguesian planes) of certain types of order. It also
leads to a possible formula for the minimum weight of the dual codes
that would be applicable for all orders.

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\bibitem{ak:book}
E.~F. Assmus, Jr. and J.~D. Key.
\newblock {\em Designs and their Codes}.
\newblock Cambridge: Cambridge University Press, 1992.
\newblock Cambridge Tracts in Mathematics, Vol.~103 (Second printing with
  corrections, 1993).

\bibitem{clakey}
K.~L. Clark and J.~D. Key.
\newblock Geometric codes over fields of odd prime power order.
\newblock {\em Congr. Numer.}, 137:177--186, 1999.

\bibitem{lincost}
Shu Lin and Daniel~J. Costello, Jr.
\newblock {\em Error Control Coding: Fundamentals and Applications}.
\newblock Prentice-Hall, 1983.
\newblock Englewood Clifts, NJ.


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