TABLE OF CONTENTS V. D. Goppa described an interesting new class of linear error-correcting codes, commonly called Goppa codes, in the early 1970s. This class of codes includes the narrow-sense BCH codes. It turned out that Goppa codes also form arguably the most interesting subclass of alternant codes, introduced by H. J. Helgert in 1974. The class of alternant codes is a large and interesting family which contains well known codes such as the BCH codes and the Goppa codes.
We encountered Reed Solomon (RS) codes in Section 8.2 as a special class of BCH codes. Recall that an RS code over F q is a BCH code over F q of length q ? 1 generated by
with a ? 1 and q ? 1 ? ? ? 2, where ? is a primitive element of F q. It is an MDS code with parameters [ q ? 1 , q ? ?, ?] (cf. Theorem 8.2.3).
Consider the case of the narrow-sense RS codes, i.e., where a = 1. In this case, there is an alternative description of the RS code that is convenient for our purpose in this chapter.
Let ? be a primitive element of the finite field F q , and let q ? 1 ? ? ? 2 . The narrow-sense q-ary RS code with generator polynomial
is equal to
| (9.1) |  |
Proof. It is easy to verify...
