Properties of syndrome distribution for blind reconstruction of cyclic codes

In the problem of blind reconstruction of channel codes, the receiver does not have the knowledge of the channel code used at the transmitter and the aim is to identify this unknown code from the received data. For the case of cyclic codes, typical blind reconstruction methods make use of some elementary properties of syndromes (remainders) of the received polynomials. The main aim of this paper is to provide a detailed analysis of the properties of the syndromes that could be useful to design more efficient blind reconstruction methods. Specifically, we prove that the syndrome distribution of the noise-free sequence can be either degenerate or uniform or restricted uniform. We also provide necessary and sufficient conditions for the syndrome distribution to be of a given type. For the noise-affected received sequence we identify additional structural properties exhibited by the syndrome distribution. Finally, we apply these results to analyze the performance of the existing blind reconstruction methods.