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

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Correspondence to Julia Lieb.

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Lieb, J. The probability of primeness for specially structured polynomial matrices over finite fields with applications to linear systems and convolutional codes. Math. Control Signals Syst. 29, 8 (2017). https://doi.org/10.1007/s00498-017-0191-z

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Keywords

  • Polynomial matrices
  • Finite fields
  • Linear systems
  • Reachability
  • Parallel connection
  • Convolutional codes