Dual codes of translation planes

J. D. Key
Department of Mathematical Sciences
Clemson University
Clemson SC 29634

Abstract

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.