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Yet another variation on minimal linear codes
Probability estimates for reachability of linear systems defined over finite fields
1. | Institut für Mathematik; Lehrstuhl für Mathematik II, Universität Würzburg, Am Hubland, 97074 Würzburg, |
2. | Institute of Mathematics, University of Würzburg, 97074 Würzburg, Germany, Germany |
References:
[1] |
J.-J. Climent, V. Herranz and C. Perea, A first approximation of concatenated convolutional codes from linear systems theory viewpoint, Linear Alg. Appl., 425 (2007), 673-699.
doi: 10.1016/j.laa.2007.03.017. |
[2] |
P. A. Fuhrmann, On controllability and observability of systems connected in parallel, IEEE Trans. Circ. Syst., 22 (1975), 57. |
[3] |
P. A. Fuhrmann and U. Helmke, The Mathematics of Networks of Linear Systems, Springer, New York, 2015.
doi: 10.1007/978-3-319-16646-9. |
[4] |
M. Garcia-Armas, S. R. Ghorpade and S. Ram, Relatively prime polynomials and nonsingular Hankel matrices over finite fields, J. Combin. Theory Ser. A, 118 (2011), 819-828.
doi: 10.1016/j.jcta.2010.11.005. |
[5] |
U. Helmke, Topology of the moduli space for reachable linear dynamical systems: The complex case, Math. Syst. Theory, 19 (1986), 155-187.
doi: 10.1007/BF01704912. |
[6] |
U. Helmke, The cohomology of moduli spaces for linear dynamical systems, Regensburger Math. Schriften, 24 (1993). |
[7] |
T. Ho and D. S. Lun, Network Coding: An Introduction, Cambridge Univ. Press, New York, 2008.
doi: 10.1017/CBO9780511754623. |
[8] |
S. Höst, Woven convolutional codes I: Encoder properties, IEEE Trans. Inf. Theory, 48 (2002), 149-161.
doi: 10.1109/18.971745. |
[9] |
A. S. Jarrah, R. Laubenbacher, B. Stigler and M. Stillman, Reverse-engineering of polynomial dynamical systems, Adv. Appl. Math., 39 (2007), 477-489.
doi: 10.1016/j.aam.2006.08.004. |
[10] |
M. Kociecky and K. M. Przyluski, On the number of controllable linear systems over a finite field, Linear Alg. Appl., 122-124 (1989), 115-122.
doi: 10.1016/0024-3795(89)90649-6. |
[11] |
J. Milnor and J. Stasheff, Characteristic Classes, Princeton Univ. Press, 1974. |
[12] |
J. A. De Reyna and R. Heyman, Counting tuples restricted by coprimality conditions, preprint, arXiv:1403.2769v1 |
[13] |
J. Rosenthal, J. M. Schumacher and E. V. York, On behaviours and convolutional codes, IEEE Trans. Inf. Theory, 42 (1996), 1881-1891.
doi: 10.1109/18.556682. |
[14] |
J. Rosenthal and E. V. York, BCH Convolutional Codes, IEEE Trans. Inf. Theory, 45 (1999), 1833-1844.
doi: 10.1109/18.782104. |
[15] |
S. Sundaram and C. Hadjicostis, Structural controllability and observability of linear systems over finite fields with applications to mult-agent systems, IEEE Trans. Autom. Control, 58 (2013), 60-73.
doi: 10.1109/TAC.2012.2204155. |
show all references
References:
[1] |
J.-J. Climent, V. Herranz and C. Perea, A first approximation of concatenated convolutional codes from linear systems theory viewpoint, Linear Alg. Appl., 425 (2007), 673-699.
doi: 10.1016/j.laa.2007.03.017. |
[2] |
P. A. Fuhrmann, On controllability and observability of systems connected in parallel, IEEE Trans. Circ. Syst., 22 (1975), 57. |
[3] |
P. A. Fuhrmann and U. Helmke, The Mathematics of Networks of Linear Systems, Springer, New York, 2015.
doi: 10.1007/978-3-319-16646-9. |
[4] |
M. Garcia-Armas, S. R. Ghorpade and S. Ram, Relatively prime polynomials and nonsingular Hankel matrices over finite fields, J. Combin. Theory Ser. A, 118 (2011), 819-828.
doi: 10.1016/j.jcta.2010.11.005. |
[5] |
U. Helmke, Topology of the moduli space for reachable linear dynamical systems: The complex case, Math. Syst. Theory, 19 (1986), 155-187.
doi: 10.1007/BF01704912. |
[6] |
U. Helmke, The cohomology of moduli spaces for linear dynamical systems, Regensburger Math. Schriften, 24 (1993). |
[7] |
T. Ho and D. S. Lun, Network Coding: An Introduction, Cambridge Univ. Press, New York, 2008.
doi: 10.1017/CBO9780511754623. |
[8] |
S. Höst, Woven convolutional codes I: Encoder properties, IEEE Trans. Inf. Theory, 48 (2002), 149-161.
doi: 10.1109/18.971745. |
[9] |
A. S. Jarrah, R. Laubenbacher, B. Stigler and M. Stillman, Reverse-engineering of polynomial dynamical systems, Adv. Appl. Math., 39 (2007), 477-489.
doi: 10.1016/j.aam.2006.08.004. |
[10] |
M. Kociecky and K. M. Przyluski, On the number of controllable linear systems over a finite field, Linear Alg. Appl., 122-124 (1989), 115-122.
doi: 10.1016/0024-3795(89)90649-6. |
[11] |
J. Milnor and J. Stasheff, Characteristic Classes, Princeton Univ. Press, 1974. |
[12] |
J. A. De Reyna and R. Heyman, Counting tuples restricted by coprimality conditions, preprint, arXiv:1403.2769v1 |
[13] |
J. Rosenthal, J. M. Schumacher and E. V. York, On behaviours and convolutional codes, IEEE Trans. Inf. Theory, 42 (1996), 1881-1891.
doi: 10.1109/18.556682. |
[14] |
J. Rosenthal and E. V. York, BCH Convolutional Codes, IEEE Trans. Inf. Theory, 45 (1999), 1833-1844.
doi: 10.1109/18.782104. |
[15] |
S. Sundaram and C. Hadjicostis, Structural controllability and observability of linear systems over finite fields with applications to mult-agent systems, IEEE Trans. Autom. Control, 58 (2013), 60-73.
doi: 10.1109/TAC.2012.2204155. |
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