Every day an extremely large amount of messages get sent over the internet, radio, WiFi, etc. These messages need to be encoded in order to protect the important data. However just like many of these systems messages can get corrupted over a certain channel due to noise, fluctuations in power or temperature etc. [2] Therefore cyclic redundancy checks (CRC), which are bits that are appended to message words to protect the data, are used to detect these errors. This is done by using CRC polynomials. An occurring undetected error is possible in any case, therefore we need to study closely the probability of an undetected error occurring, that is the probability that the received, erroneous data is accepted as the transmitted message. The code words that are the results of adding cyclic redunancy checks (CRC) can also be combined with an error-correcting codes like BCH codes. As shown in [3] combining CRC polynomials with error-correcting can reduce the previously mentioned probability, but one CRC polynomial can perform better. In this thesis these CRC polynomials are studied in combination with certain different codes and are evaluated based on their performance with respects to their undetected error probability. Two examples of codes are given, in the first example a CRC polynomial is found that does not perform well in the system without using an error-correcting code, but performs the best in the system with an error-correcting code. An analysis is done which shows that by looking at consecutive roots of the generator polynomial, optimal CRC polynomials can be found. The second example uses a different code and finds another optimal CRC polynomial