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|Title:||Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields|
|Keywords:||Feedback Shift Registers|
|Publisher:||ACADEMIC PRESS INC ELSEVIER SCIENCE|
|Citation:||FINITE FIELDS AND THEIR APPLICATIONS,17(5)461-472|
|Abstract:||We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed. (C) 2011 Elsevier Inc. All rights reserved.|
|Appears in Collections:||Article|
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