The probability of primeness for specially structured polynomial matrices over finite fields with applications to linear systems and convolutional codes

We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many non-singular polynomial matrices are mutually left coprime. These results are used to estimate the number of reachable and observable linear systems as well as the number of non-catastrophic convolutional codes. Moreover, we are able to achieve an asymptotic formula for the probability that a parallel connected linear system is reachable.