CLASSIFICATION OF GENERALIZED BCH CODES AND SOME PROPERTIES OF GOPPA CODES.

Using a common parity checking matrix, various codes which can be viewed as generalizations of BCH codes are systematically consolidated and their relations are clarified. As a result, it is found that the ″generalized Srivastava code″ and Goppa code are subclasses of H. J. Helgert's ″generalized Srivastava code″ and which Helgert called ″an interesting new class of code″ is, in effect, a subclass of Goppa codes. New and interesting properties of Goppa codes are elucidated. That is, if g(z) is an arbitrary polynomial of order t//0 with elements of GF(q**m) as its coefficients, the Goppa code on GF(q) that has a Goppa polynomial q**q(z) has code length less than equivalent to q**m, number of check symbols less than equivalent to (q minus 1)mt//0 and minimum intercode distance greater than equivalent to qt//0 plus 1. A particularly interesting property is that (1) if q does not equal 2, the code length can be up to q**m when g(z) does not have elements of GF(q**m) as its roots and (2) if q equals 2, g(z) can be an arbitrary polynomial.