An exact evaluation of the probability of undetected error for certain shortened binary CRC codes

The authors give a computationally efficient algorithm for computing the probability of undetected error for a class of cyclic codes whose generator polynomial is equal to (x+1) times a primitive irreducible polynomial. This class contains three CRC (cyclic redundancy check) codes that have been adopted as international standards. The algorithm was used to compute the performance of a number of codes at various shortened block lengths, often with surprising results. It is suggested that, when dealing with shortened block lengths, one should choose a primitive polynomial with many rather than few nonzero coefficients in order to produce a good code.<>