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

TL;DR

This paper derives asymptotic probabilities for the primeness and coprimeness of structured polynomial matrices over finite fields, with applications to analyzing properties of linear systems and convolutional codes.

Contribution

It provides new asymptotic formulas for the likelihood of polynomial matrices being prime or coprime, aiding in system and code property estimations.

Findings

Asymptotic formula for the probability of polynomial matrices being right or left prime.

Estimation of the number of reachable and observable linear systems.

Probability estimates for non-catastrophic convolutional codes.

Abstract

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 nonsingular 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.