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.