The McEliece cryptographic system is one of the finalists of the second round of the NIST Post-Quantum Cryptography Standardization Process. As such, it will likely be used in many instances to ensure secure communications over the Internet when quantum computers become practical. The selection of the new system, which will replace the widely used RSA cryptographic system, should be announced by 2025. The McEliece cryptographic system is code-based, which means that it uses an error-correcting code as its basis, typically a Goppa code; it is the random insertion of errors in the original message that guarantees the security of the system. Presumably, only the legitimate user is capable of correcting those errors, thereby retrieving the message for which it is intended. As a result, an efficient decoding algorithm is required. In this work, we study two decoding algorithms for Goppa codes, namely, the Euclidean and Patterson, implement them in Magma, and compare their performances considering several code parameters. Lastly, a modification to Patterson’s algorithm is proposed to handle certain cases where the Goppa polynomial is reducible.