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The weight distribution of the self-dual $[128,64]$ polarity design code
Constructions and bounds for mixed-dimension subspace codes
1. | Department of Information Science and Electronics Engineering, Zhejiang University, 38 Zheda Road, Hangzhou 310027 |
2. | Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth |
3. | Institut für Mathematik, Universität Bayreuth, D-95440 Bayreuth |
References:
[1] |
R. Ahlswede and H. Aydinian, On error control codes for random network coding, in IEEE Workshop Network Coding Theory Appl., 2009, 68-73.
doi: 10.1109/NETCOD.2009.5191396. |
[2] |
C. Bachoc, A. Passuello, and F. Vallentin, Bounds for projective codes from semidefinite programming, Adv. Math. Commun., 7 (2013), 127-145.
doi: 10.3934/amc.2013.7.127. |
[3] |
J. De Beule and L. Storme, Current Research Topics in Galois Geometry, Nova Science Publishers, 2011. |
[4] |
A. Beutelspacher, Partial spreads in finite projective spaces and partial designs, Math. Z., 145 (1975), 211-230. Corrigendum, ibid., 147 (1976), 303.
doi: 10.1007/BF01215286. |
[5] |
M. Braun and J. Reichelt, $q$-analogs of packing designs, J. Combin. Des., 22 (2014), 306-321.
doi: 10.1002/jcd.21376. |
[6] |
P. J. Cameron and J. H. van Lint, Graphs, Codes and Designs, Cambridge Univ. Press, 1980. A revised edition of these notes is [7]. |
[7] |
P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and their Links, Cambridge Univ. Press, 1991. Revised edition of [6].
doi: 10.1017/CBO9780511623714. |
[8] |
A. Cossidente, F. Pavese and L. Storme, Optimal subspace codes in $PG(4,q)$,, in preparation., ().
|
[9] |
T. Czerwinski and D. Oakden, The translation planes of order twenty-five, J. Combin. Theory Ser. A, 59 (1992), 193-217.
doi: 10.1016/0097-3165(92)90065-3. |
[10] |
P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[11] |
P. Dembowski, Finite Geometries, Springer-Verlag, 1968.
doi: 10.1007/978-3-642-62012-6. |
[12] |
U. Dempwolff, Translation planes of order 27, Des. Codes Cryptogr., 4 (1994), 105-121. Erratum, ibid., 5 (1995), 81.
doi: 10.1007/BF01578865. |
[13] |
U. Dempwolff and A. Reifart, The classification of the translation planes of order 16, I, Geom. Dedicata, 15 (1983), 137-153.
doi: 10.1007/BF00147760. |
[14] |
J. Eisfeld and L. Storme, (Partial) $t$-Spreads and Minimal $t$-Covers in Finite Projective Spaces, Lecture notes, Ghent Univ., 2000. |
[15] |
T. Etzion, Problems on $q$-analogs in coding theory,, preprint, ().
|
[16] |
T. Etzion and N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017. Erratum, ibid., 59 (2013), 4730.
doi: 10.1109/TIT.2012.2220119. |
[17] |
T. Etzion and L. Storme, Galois geometries and coding theory, Des. Codes Cryptogr., 78 (2016), 311-350.
doi: 10.1007/s10623-015-0156-5. |
[18] |
T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[19] |
T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Adv. Math. Commun., 3 (2009), 363-383.
doi: 10.3934/amc.2009.3.363. |
[20] |
T. Feulner, Canonical forms and automorphisms in the projective space,, preprint, ().
|
[21] |
T. Feulner, Eine kanonische Form zur Darstellung äquivalenter Codes - Computergestützte Berechnung und ihre Anwendung in der Codierungstheorie, Kryptographie und Geometrie, Ph.D thesis, Univ. Bayreuth, 2014. Available online at https://epub.uni-bayreuth.de/42/ |
[22] |
E. M. Gabidulin and M. Bossert, Algebraic codes for network coding, Probl. Inform. Transm., 45 (2009), 343-356.
doi: 10.1134/S003294600904005X. |
[23] |
J. Galambos and I. Simonelli, Bonferroni-Type Inequalities with Applications, Springer-Verlag, 1996. |
[24] |
N. A. Gordon, R. Shaw and L. H. Soicher, Classification of partial spreads in $\PG(4,2)$,, available online at , ().
|
[25] |
X. Guang and Z. Zhang, Linear Network Error Correction Coding, Springer-Verlag, 2014.
doi: 10.1007/978-1-4939-0588-1. |
[26] |
M. Hall, Jr., J. D. Swift and R. J. Walker, Uniqueness of the projective plane of order eight, Math. Comp., 10 (1956), 186-194.
doi: 10.2307/2001913. |
[27] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes,, preprint, ().
|
[28] |
J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford Univ. Press, 1985. |
[29] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Univ. Press, 1998. |
[30] |
J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford Univ. Press, 1991.
doi: 10.1007/978-1-4471-6790-7. |
[31] |
T. Honold and M. Kiermaier, On putative $q$-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday (eds. T. Hagen, F. Rupp and J. Scheurle), World Scientific, 2016, 141-175. Available online at arXiv:1504.06688
doi: 10.1142/9789814699877_0008. |
[32] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, in 11th Int. Conf. Finite Fields Appl., 2013, Magdeburg, 2015, 157-176.
doi: 10.1090/conm/632/12627. |
[33] |
T. Honold, M. Kiermaier and S. Kurz, Classification of large partial plane spreads in $\PG(6,\mathbbF_2)$ and related combinatorial objects, submitted., Available online at , ().
|
[34] |
N. L. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes, CRC Press, 2007.
doi: 10.1201/9781420011142. |
[35] |
D. J. Kleitman, On an extremal property of antichains in partial orders. The LYM property and some of its implications and applications, in Combinatorics, Springer, 1975, 277-290.
doi: 10.1007/978-94-010-1826-5_14. |
[36] |
R. Koetter and F. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[37] |
J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992. |
[38] |
H. Liu and T. Honold, Poster: A new approach to the main problem of subspace coding, in 9th Int. Conf. Commun. Netw. China (ChinaCom 2014), 2014, 676-677.
doi: 10.1109/CHINACOM.2014.7054392. |
[39] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, Amsterdam, 1977. |
[40] |
R. Mathon and G. F. Royle, The translation planes of order 49, Des. Codes Cryptogr., 5 (1995), 57-72.
doi: 10.1007/BF01388504. |
[41] |
B. D. McKay and A. Piperno, Practical graph isomorphism II, J. Symb. Comput., 60 (2014), 94-112.
doi: 10.1016/j.jsc.2013.09.003. |
[42] |
G. E. Moorhouse, Two-graphs and skew two-graphs in finite geometries, Linear Algebra Appl., 226-228 (1995), 529-551.
doi: 10.1016/0024-3795(95)00242-J. |
[43] |
S. Niskanen and P. R. J. Östergård, Cliquer User's Guide, Version 1.0, Commun. Lab., Helsinki Univ. Techn., Finland, Tech. Rep. T48, 2003. |
[44] |
D. Silva, F. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[45] |
D. Silva, F. Kschischang and R. Koetter, Communication over finite-field matrix channels, IEEE Trans. Inform. Theory, 56 (2010), 1296-1306.
doi: 10.1109/TIT.2009.2039167. |
[46] |
A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.
doi: 10.3934/amc.2013.7.147. |
[47] |
R. W. Yeung and N. Cai, Network error correction, part I: Basic concepts and upper bounds, Commun. Inform. Syst., 6 (2006), 19-35. |
[48] |
R. W. Yeung and N. Cai, Network error correction, part II: Lower bounds, Commun. Inform. Syst., 6 (2006), 37-54. |
show all references
References:
[1] |
R. Ahlswede and H. Aydinian, On error control codes for random network coding, in IEEE Workshop Network Coding Theory Appl., 2009, 68-73.
doi: 10.1109/NETCOD.2009.5191396. |
[2] |
C. Bachoc, A. Passuello, and F. Vallentin, Bounds for projective codes from semidefinite programming, Adv. Math. Commun., 7 (2013), 127-145.
doi: 10.3934/amc.2013.7.127. |
[3] |
J. De Beule and L. Storme, Current Research Topics in Galois Geometry, Nova Science Publishers, 2011. |
[4] |
A. Beutelspacher, Partial spreads in finite projective spaces and partial designs, Math. Z., 145 (1975), 211-230. Corrigendum, ibid., 147 (1976), 303.
doi: 10.1007/BF01215286. |
[5] |
M. Braun and J. Reichelt, $q$-analogs of packing designs, J. Combin. Des., 22 (2014), 306-321.
doi: 10.1002/jcd.21376. |
[6] |
P. J. Cameron and J. H. van Lint, Graphs, Codes and Designs, Cambridge Univ. Press, 1980. A revised edition of these notes is [7]. |
[7] |
P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and their Links, Cambridge Univ. Press, 1991. Revised edition of [6].
doi: 10.1017/CBO9780511623714. |
[8] |
A. Cossidente, F. Pavese and L. Storme, Optimal subspace codes in $PG(4,q)$,, in preparation., ().
|
[9] |
T. Czerwinski and D. Oakden, The translation planes of order twenty-five, J. Combin. Theory Ser. A, 59 (1992), 193-217.
doi: 10.1016/0097-3165(92)90065-3. |
[10] |
P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[11] |
P. Dembowski, Finite Geometries, Springer-Verlag, 1968.
doi: 10.1007/978-3-642-62012-6. |
[12] |
U. Dempwolff, Translation planes of order 27, Des. Codes Cryptogr., 4 (1994), 105-121. Erratum, ibid., 5 (1995), 81.
doi: 10.1007/BF01578865. |
[13] |
U. Dempwolff and A. Reifart, The classification of the translation planes of order 16, I, Geom. Dedicata, 15 (1983), 137-153.
doi: 10.1007/BF00147760. |
[14] |
J. Eisfeld and L. Storme, (Partial) $t$-Spreads and Minimal $t$-Covers in Finite Projective Spaces, Lecture notes, Ghent Univ., 2000. |
[15] |
T. Etzion, Problems on $q$-analogs in coding theory,, preprint, ().
|
[16] |
T. Etzion and N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017. Erratum, ibid., 59 (2013), 4730.
doi: 10.1109/TIT.2012.2220119. |
[17] |
T. Etzion and L. Storme, Galois geometries and coding theory, Des. Codes Cryptogr., 78 (2016), 311-350.
doi: 10.1007/s10623-015-0156-5. |
[18] |
T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[19] |
T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Adv. Math. Commun., 3 (2009), 363-383.
doi: 10.3934/amc.2009.3.363. |
[20] |
T. Feulner, Canonical forms and automorphisms in the projective space,, preprint, ().
|
[21] |
T. Feulner, Eine kanonische Form zur Darstellung äquivalenter Codes - Computergestützte Berechnung und ihre Anwendung in der Codierungstheorie, Kryptographie und Geometrie, Ph.D thesis, Univ. Bayreuth, 2014. Available online at https://epub.uni-bayreuth.de/42/ |
[22] |
E. M. Gabidulin and M. Bossert, Algebraic codes for network coding, Probl. Inform. Transm., 45 (2009), 343-356.
doi: 10.1134/S003294600904005X. |
[23] |
J. Galambos and I. Simonelli, Bonferroni-Type Inequalities with Applications, Springer-Verlag, 1996. |
[24] |
N. A. Gordon, R. Shaw and L. H. Soicher, Classification of partial spreads in $\PG(4,2)$,, available online at , ().
|
[25] |
X. Guang and Z. Zhang, Linear Network Error Correction Coding, Springer-Verlag, 2014.
doi: 10.1007/978-1-4939-0588-1. |
[26] |
M. Hall, Jr., J. D. Swift and R. J. Walker, Uniqueness of the projective plane of order eight, Math. Comp., 10 (1956), 186-194.
doi: 10.2307/2001913. |
[27] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes,, preprint, ().
|
[28] |
J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford Univ. Press, 1985. |
[29] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Univ. Press, 1998. |
[30] |
J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford Univ. Press, 1991.
doi: 10.1007/978-1-4471-6790-7. |
[31] |
T. Honold and M. Kiermaier, On putative $q$-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday (eds. T. Hagen, F. Rupp and J. Scheurle), World Scientific, 2016, 141-175. Available online at arXiv:1504.06688
doi: 10.1142/9789814699877_0008. |
[32] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, in 11th Int. Conf. Finite Fields Appl., 2013, Magdeburg, 2015, 157-176.
doi: 10.1090/conm/632/12627. |
[33] |
T. Honold, M. Kiermaier and S. Kurz, Classification of large partial plane spreads in $\PG(6,\mathbbF_2)$ and related combinatorial objects, submitted., Available online at , ().
|
[34] |
N. L. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes, CRC Press, 2007.
doi: 10.1201/9781420011142. |
[35] |
D. J. Kleitman, On an extremal property of antichains in partial orders. The LYM property and some of its implications and applications, in Combinatorics, Springer, 1975, 277-290.
doi: 10.1007/978-94-010-1826-5_14. |
[36] |
R. Koetter and F. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[37] |
J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992. |
[38] |
H. Liu and T. Honold, Poster: A new approach to the main problem of subspace coding, in 9th Int. Conf. Commun. Netw. China (ChinaCom 2014), 2014, 676-677.
doi: 10.1109/CHINACOM.2014.7054392. |
[39] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, Amsterdam, 1977. |
[40] |
R. Mathon and G. F. Royle, The translation planes of order 49, Des. Codes Cryptogr., 5 (1995), 57-72.
doi: 10.1007/BF01388504. |
[41] |
B. D. McKay and A. Piperno, Practical graph isomorphism II, J. Symb. Comput., 60 (2014), 94-112.
doi: 10.1016/j.jsc.2013.09.003. |
[42] |
G. E. Moorhouse, Two-graphs and skew two-graphs in finite geometries, Linear Algebra Appl., 226-228 (1995), 529-551.
doi: 10.1016/0024-3795(95)00242-J. |
[43] |
S. Niskanen and P. R. J. Östergård, Cliquer User's Guide, Version 1.0, Commun. Lab., Helsinki Univ. Techn., Finland, Tech. Rep. T48, 2003. |
[44] |
D. Silva, F. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[45] |
D. Silva, F. Kschischang and R. Koetter, Communication over finite-field matrix channels, IEEE Trans. Inform. Theory, 56 (2010), 1296-1306.
doi: 10.1109/TIT.2009.2039167. |
[46] |
A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.
doi: 10.3934/amc.2013.7.147. |
[47] |
R. W. Yeung and N. Cai, Network error correction, part I: Basic concepts and upper bounds, Commun. Inform. Syst., 6 (2006), 19-35. |
[48] |
R. W. Yeung and N. Cai, Network error correction, part II: Lower bounds, Commun. Inform. Syst., 6 (2006), 37-54. |
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