Encoding Cyclic Redundancy Checked Sequences

The undetected error probability of cyclic redundancy checks (CRC) combined with an error-correcting code in a concatenated coding scheme is considered. It is shown that this probability depends very much on the encoding of the cyclic redundancy checked sequences into codewords in the error-correcting code. Furthermore, it is proved that the average undetected error probability of the concatenated code over all linear encoders of the error-correcting code is about 1/2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> of that of the error-correcting code, where ℓ is the degree of the CRC polynomial. In particular, for any CRC polynomial of degree ℓ, there exists an encoding function of the error-correcting code such that the undetected error probability of the concatenated scheme is at most 1/2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> of that of the error-correcting code.