The Probability of Primeness for Specially Structured Polynomial Matrices over Finite Fields with Applications to Linear Systems and Convolutional Codes

摘要

We calculate the probability that random polynomial matrices over a finitefield with certain structures are right prime or left prime, respectively. Inparticular, we give an asymptotic formula for the probability that finitelymany nonsingular polynomial matrices are mutually left coprime. These resultsare used to estimate the number of reachable and observable linear systems aswell as the number of non-catastrophic convolutional codes. Moreover, we areable to achieve an asymptotic formula for the probability that a parallelconnected linear system is reachable.