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Polar codes for distributed hierarchical source coding
Derived and residual subspace designs
1. | Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany |
2. | Institut für Informatik, Universität Bayreuth, D-95440 Bayreuth, Germany |
  As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no $q$-analog of the large Witt design.
References:
[1] |
S. Ajoodani-Namini and G. B. Khosrovashahi, More on halving the complete designs, Discrete Math., 135 (1994), 29-37.
doi: 10.1016/0012-365X(93)E0096-M. |
[2] |
M. Braun, Some new designs over finite fields, Bayreuther Math. Schr., 74 (2005), 58-68. |
[3] |
M. Braun, T. Etzion, P. R. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems, preprint, arXiv:1304.1462 |
[4] |
M. Braun, A. Kerber and R. Laue, Systematic construction of $q$-analogs of designs, Des. Codes Cryptogr., 34 (2005), 55-70.
doi: 10.1007/s10623-003-4194-z. |
[5] |
M. Braun, A. Kohnert, P. R. Östergård and A. Wassermann, Large sets of $t$-designs over finite fields, J. Combin. Theory Ser. A, 124 (2014), 195-202.
doi: 10.1016/j.jcta.2014.01.008. |
[6] |
S. Braun, Algorithmen zur computerunterstützten Berechnung von $q$-Analoga kombinatorischer Designs, diploma thesis, Universität Bayreuth, 2009. |
[7] |
S. Braun, Construction of $q$-analogs of combinatorial designs, presentation at the conference Algebraic Combinatorics and Applications (ALCOMA10), Thurnau, Germany, 2010. |
[8] |
P. J. Cameron, Generalization of Fisher's inequality to fields with more than one element, in Proc. British Combinat. Conf. 1973, Cambridge University Press, 1974, 9-13.
doi: 10.1017/CBO9780511662072.003. |
[9] |
H. Cohn, Projective geometry over $\mathbb F_1$ and the Gaussian binomial coefficients, Amer. Math. Monthly, 111 (2004), 487-495.
doi: 10.2307/4145067. |
[10] |
L. M. H. E. Driessen, $t$-designs, $t \ge 3$, technical report, Technische Universiteit Eindhoven, 1978. |
[11] |
A. Fazeli, S. Lovett and A. Vardy, Nontrivial $t$-designs over finite fields exist for all $t$, preprint, arXiv:1306.2088 |
[12] |
J. Goldman and G.-C. Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions, Stud. Appl. Math., 49 (1970), 239-258. |
[13] |
T. Itoh, A new family of $2$-designs over $GF(q)$ admitting $SL_m(q^l)$, Geom. Dedicata, 69 (1998), 261-286.
doi: 10.1023/A:1005057610394. |
[14] |
M. Kiermaier and M. O. Pavčević, Intersection numbers for subspace designs, J. Combin. Des., to appear: doi:10.1002/jcd.21403
doi: 10.1002/jcd.21403. |
[15] |
D. C. van Leijenhorst, Orbits on the projective line, J. Combin. Theory Ser. A, 31 (1981), 146-154.
doi: 10.1016/0097-3165(81)90011-X. |
[16] |
M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small $2$-designs over GF$(q)$, J. Combin. Des., 3 (1995), 61-77.
doi: 10.1002/jcd.3180030108. |
[17] |
M. Schwartz and T. Etzion, Codes and anticodes in the Grassman graph, J. Combin. Theory Ser. A, 97 (2002), 27-42.
doi: 10.1006/jcta.2001.3188. |
[18] |
H. Suzuki, Five days introduction to the theory of designs, 1989, available online at http://subsite.icu.ac.jp/people/hsuzuki/lecturenote/designtheory.pdf |
[19] |
H. Suzuki, On the inequalities of $t$-designs over a finite field, European J. Combin., 11 (1990), 601-607.
doi: 10.1016/S0195-6698(13)80045-5. |
[20] |
H. Suzuki, $2$-designs over $GF(2^m)$, Graphs Combin., 6 (1990), 293-296.
doi: 10.1007/BF01787580. |
[21] |
H. Suzuki, $2$-designs over $GF(q)$, Graphs Combin., 8 (1992), 381-389.
doi: 10.1007/BF02351594. |
[22] |
L. Teirlinck, Non-trivial $t$-designs without repeated blocks exist for all $t$, Discrete Math., 65 (1987), 301-311.
doi: 10.1016/0012-365X(87)90061-6. |
[23] |
S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242.
doi: 10.1007/BF00150939. |
[24] |
J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d'Algébre Supérieure, Librairie Gauthiers-Villars, Paris, 1957, 261-289. |
[25] |
Tran Van Trung, On the construction of $t$-designs and the existence of some new infinite families of simple $5$-designs, Arch. Math. (Basel), 47 (1986), 187-192.
doi: 10.1007/BF01193690. |
show all references
References:
[1] |
S. Ajoodani-Namini and G. B. Khosrovashahi, More on halving the complete designs, Discrete Math., 135 (1994), 29-37.
doi: 10.1016/0012-365X(93)E0096-M. |
[2] |
M. Braun, Some new designs over finite fields, Bayreuther Math. Schr., 74 (2005), 58-68. |
[3] |
M. Braun, T. Etzion, P. R. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems, preprint, arXiv:1304.1462 |
[4] |
M. Braun, A. Kerber and R. Laue, Systematic construction of $q$-analogs of designs, Des. Codes Cryptogr., 34 (2005), 55-70.
doi: 10.1007/s10623-003-4194-z. |
[5] |
M. Braun, A. Kohnert, P. R. Östergård and A. Wassermann, Large sets of $t$-designs over finite fields, J. Combin. Theory Ser. A, 124 (2014), 195-202.
doi: 10.1016/j.jcta.2014.01.008. |
[6] |
S. Braun, Algorithmen zur computerunterstützten Berechnung von $q$-Analoga kombinatorischer Designs, diploma thesis, Universität Bayreuth, 2009. |
[7] |
S. Braun, Construction of $q$-analogs of combinatorial designs, presentation at the conference Algebraic Combinatorics and Applications (ALCOMA10), Thurnau, Germany, 2010. |
[8] |
P. J. Cameron, Generalization of Fisher's inequality to fields with more than one element, in Proc. British Combinat. Conf. 1973, Cambridge University Press, 1974, 9-13.
doi: 10.1017/CBO9780511662072.003. |
[9] |
H. Cohn, Projective geometry over $\mathbb F_1$ and the Gaussian binomial coefficients, Amer. Math. Monthly, 111 (2004), 487-495.
doi: 10.2307/4145067. |
[10] |
L. M. H. E. Driessen, $t$-designs, $t \ge 3$, technical report, Technische Universiteit Eindhoven, 1978. |
[11] |
A. Fazeli, S. Lovett and A. Vardy, Nontrivial $t$-designs over finite fields exist for all $t$, preprint, arXiv:1306.2088 |
[12] |
J. Goldman and G.-C. Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions, Stud. Appl. Math., 49 (1970), 239-258. |
[13] |
T. Itoh, A new family of $2$-designs over $GF(q)$ admitting $SL_m(q^l)$, Geom. Dedicata, 69 (1998), 261-286.
doi: 10.1023/A:1005057610394. |
[14] |
M. Kiermaier and M. O. Pavčević, Intersection numbers for subspace designs, J. Combin. Des., to appear: doi:10.1002/jcd.21403
doi: 10.1002/jcd.21403. |
[15] |
D. C. van Leijenhorst, Orbits on the projective line, J. Combin. Theory Ser. A, 31 (1981), 146-154.
doi: 10.1016/0097-3165(81)90011-X. |
[16] |
M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small $2$-designs over GF$(q)$, J. Combin. Des., 3 (1995), 61-77.
doi: 10.1002/jcd.3180030108. |
[17] |
M. Schwartz and T. Etzion, Codes and anticodes in the Grassman graph, J. Combin. Theory Ser. A, 97 (2002), 27-42.
doi: 10.1006/jcta.2001.3188. |
[18] |
H. Suzuki, Five days introduction to the theory of designs, 1989, available online at http://subsite.icu.ac.jp/people/hsuzuki/lecturenote/designtheory.pdf |
[19] |
H. Suzuki, On the inequalities of $t$-designs over a finite field, European J. Combin., 11 (1990), 601-607.
doi: 10.1016/S0195-6698(13)80045-5. |
[20] |
H. Suzuki, $2$-designs over $GF(2^m)$, Graphs Combin., 6 (1990), 293-296.
doi: 10.1007/BF01787580. |
[21] |
H. Suzuki, $2$-designs over $GF(q)$, Graphs Combin., 8 (1992), 381-389.
doi: 10.1007/BF02351594. |
[22] |
L. Teirlinck, Non-trivial $t$-designs without repeated blocks exist for all $t$, Discrete Math., 65 (1987), 301-311.
doi: 10.1016/0012-365X(87)90061-6. |
[23] |
S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242.
doi: 10.1007/BF00150939. |
[24] |
J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d'Algébre Supérieure, Librairie Gauthiers-Villars, Paris, 1957, 261-289. |
[25] |
Tran Van Trung, On the construction of $t$-designs and the existence of some new infinite families of simple $5$-designs, Arch. Math. (Basel), 47 (1986), 187-192.
doi: 10.1007/BF01193690. |
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