The new theorems of solving lower bound on the minimum distance of Goppa codes

Abstract

The author shows the new proof of Blahut's theorem (1979, 1983) by use of the Z-transformation. By applying the finite field DFT and Blahut's theorem, they present two theorems which can be used to solve the lower bound on the minimum distance of Goppa codes. Given the generator matrix G or the parity check matrix H of a Goppa code, it is very convenient to get the lower bounds on the minimum distance of the Goppa code by use of these theorems. These lower bounds are more effective than the known lower bound of Mac Williams (1977), sometimes it is not as effective as the lower bound given by Loeloeian and Conan (1987), however using these theorems to solve the lower bound is much simpler than using the L-C bound. Some examples are illustrated employing the two theorems, the known bound and the L-C bound, respectively.<>