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 {\bf Permutation Decoding}


J. D. Key\\ Department of Mathematical Sciences\\
Clemson University\\ Clemson SC 29634
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Permutation decoding was first developed by Jessie MacWilliams in the
early 60's.  It can be used when a code has sufficiently many
automorphisms to ensure the existence of set of automorphisms that
satisfy certain conditions: a PD-set for a code is a set S of
automorphisms of the code which is such that, if the code can correct
t errors, then every possible error vector of weight t or less can be
moved by some member of S out of the information positions. That such
a set will fully use the error-correction potential of the code
follows from an early result which we will prove, but that such a set
exists at all is clearly not always true. There is a bound on the
minimum size that the set S may have, which we will quote.

This talk will mostly simply explain the method, and some ways of
finding such sets for some particular codes with large groups, for
example cyclic codes, or those with some of the classical groups
acting.  A short Magma demonstration of the decoding might be
included, finding some PD-sets for some binary codes of length 28 with
a unitary group acting, and some 5-ary cyclic codes of length 31 with
the projective general linear group acting, and showing how they work.


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\bibitem{huffman}
W.~Cary Huffman.
\newblock Codes and groups.
\newblock In V.~S. Pless and W.~C. Huffman, editors, {\em Handbook of {C}oding
  {T}heory}, pages 1345--1440. Amsterdam: Elsevier, 1998.
\newblock Volume~2, Part~2, Chapter~17.


\bibitem{macslo}
F.~J. MacWilliams and N.~J.~A. Sloane.
\newblock {\em The Theory of Error-Correcting Codes}.
\newblock Amsterdam: North-Holland, 1983.

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