Quintic Irreducible Binary Goppa Codes of Length 32

Goppa codes are a class of linear codes and have a well defined algebraic structure. They are considered to be very close to random codes and form the basis of the Mc Eliece public key cryptosystem. Their true minimum distance and dimension are still not known. Neither do we know which choices of Goppa polynomials give rise to codes with better minimum distance. In this book we give an exact count of quintic irreducible Goppa codes of length 32 and their extended versions. We also present an instance of the problem of true minimum distance and choice of a Goppa polynomial. Apart from this, we give an example to show that the well known sufficient condition for two Goppa codes to be equivalent is not a necessary condition. We point out that it is possible to get such an example from among the degenerate codes but we show that the example we give is taken from the non-degenerate codes. We then give a count of degenerate and non degenerate codes.