In McEliece cryptosystem, G matrix is scrambled using S and P so that scrambled G matrix is G' = SGP. Here G is the generator matrix of a linear code and after scrambling it is converted into another matrix G' of the same size. How can we ensure that G' is another possible G matrix of the code with same distance properties? Is there any proof for that? If G' satisfies all the properties of G, will this hold any linear code other than Goppa code also?
c = mSGP + e
Here mS will give you another k element vector which forms the message for the permuted version of the G matrix, GP. Now my question is whether the permuted G matrix results in another valid G matrix. Are there any properties to be satisfied by P matrix?