This article presents an algorithm for finding enumerators of the second degree and higher for a generalized (L, G)-code. This algorithm allows you to increase the speed of calculating enumerators, in contrast to the exhaustive search algorithm. This algorithm can be used to build a modern version of the McEliece cryptosystem. The above solution is based on using the method of representing an element of the Galois field through a function containing coefficients from a field of smaller size. This article also presents the results of a comparison of the modern McEliece cryptosystem built on Goppa codes and generalized (L, G)-codes