November 2018, 12(4): 817-839. doi: 10.3934/amc.2018048

Binary subspace codes in small ambient spaces

1. 

Department of Communications and Networking, Aalto University, Helsinki, Finland

2. 

Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany

Received  April 2018 Revised  June 2018 Published  September 2018

Fund Project: The authors were supported by the DFG project "Ganzzahlige Optimierungsmodelle für Subspace Codes und endliche Geometrie" (DFG grants KU 2430/3-1, WA 1666/9-1)

Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for projective dimensions of at most $ 7 $, i.e., affine dimensions of at most $ 8 $. We obtain several improvements of the bounds and perform two classifications of optimal subspace codes, which are unknown so far in the literature.

Citation: Daniel Heinlein, Sascha Kurz. Binary subspace codes in small ambient spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 817-839. doi: 10.3934/amc.2018048
References:
[1]

R. Ahlswede and H. Aydinian, On error control codes for random network coding, in Network Coding, Theory, and Applications, 2009. NetCod'09. Workshop on, IEEE, 2009, 68-73. doi: 10.1109/NETCOD.2009.5191396.

[2]

C. BachocA. Passuello and F. Vallentin, Bounds for projective codes from semidefinite programming, Advances in Mathematics of Communications, 7 (2013), 127-145. doi: 10.3934/amc.2013.7.127.

[3]

M. BraunP. R. Östergård and A. Wassermann, New lower bounds for binary constant-dimension subspace codes, Experimental Mathematics, 27 (2018), 179-183. doi: 10.1080/10586458.2016.1239145.

[4]

M. Braun and J. Reichelt, $ q $-analogs of packing designs, Journal of Combinatorial Designs, 22 (2014), 306-321. doi: 10.1002/jcd.21376.

[5]

N. Cai and R. W. Yeung, Network error correction, Ⅱ: Lower bounds, Communications in Information & Systems, 6 (2006), 37-54. doi: 10.4310/CIS.2006.v6.n1.a3.

[6]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, Journal of Combinatorial Theory, Series A, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.

[7]

P. Dembowski, Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44 Springer-Verlag, Berlin-New York, 1968.

[8]

U. Dempwolff and A. Reifart, The classification of the translation planes of order 16, I, Geometriae Dedicata, 15 (1983), 137-153. doi: 10.1007/BF00147760.

[9]

T. EtzionE. GorlaA. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Transactions on Information Theory, 62 (2016), 1616-1630. doi: 10.1109/TIT.2016.2522971.

[10]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Transactions on Information Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.

[11]

T. Etzion and N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Transactions on Information Theory, 59 (2013), 1004-1017. doi: 10.1109/TIT.2012.2220119.

[12]

T. Etzion and L. Storme, Galois geometries and coding theory, Designs, Codes and Cryptography, 78 (2016), 311-350. doi: 10.1007/s10623-015-0156-5.

[13]

T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Advances in Mathematics of Communications, 3 (2009), 363-383. doi: 10.3934/amc.2009.3.363.

[14]

T. Feulner, Eine kanonische Form zur Darstellung äquivalenter Codes: Computergestützte Berechnung und ihre Anwendung in der Codierungstheorie, Kryptographie und Geometrie, PhD thesis, University of Bayreuth, Bayreuth, 2014.

[15]

E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.

[16]

M. Greferath, M. O. Pavčević, N. Silberstein and M. Á. Vázquez-Castro, Network Coding and Subspace Designs, Springer, 2018. doi: 10.1007/978-3-319-70293-3.

[17]

M. Hall JrJ. D. Swift and R. J. Walker, Uniqueness of the projective plane of order eight, Mathematical Tables and Other Aids to Computation, 10 (1956), 186-194. doi: 10.2307/2001913.

[18]

D. Heinlein, T. Honold, M. Kiermaier, S. Kurz and A. Wassermann, Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6 to appear in Designs, Codes and Cryptography, https://link.springer.com/article/10.1007%2Fs10623-018-0544-8. doi: 10.1007/s10623-018-0544-8.

[19]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864.

[20]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, A subspace code of size $333 $ in the setting of a binary $ q $-analog of the Fano plane, preprint, arXiv: 1708.06224.

[21]

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, World Scientific, 2016,141-175.

[22]

T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4, in Topics in Finite Fields, vol. 632 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2015,157-176. doi: 10.1090/conm/632/12627.

[23]

T. Honold, M. Kiermaier and S. Kurz, Classification of large partial plane spreads in $ {P}{G}(6, 2) $ and related combinatorial objects, to appear in Journal of Geometry. arXiv: 1606.07655.

[24]

T. HonoldM. Kiermaier and S. Kurz, Constructions and bounds for mixed-dimension subspace codes, Advances in Mathematics of Communications, 10 (2016), 649-682. doi: 10.3934/amc.2016033.

[25]

T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, Springer, 2018,131-170.

[26]

N. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes, CRC Press, 2007. doi: 10.1201/9781420011142.

[27]

M. Kiermaier and S. Kurz, An improvement of the Johnson bound for subspace codes, preprint, arXiv: 1707.00650.

[28]

M. KiermaierS. Kurz and A. Wassermann, The order of the automorphism group of a binary $ q $-analog of the Fano plane is at most two, Designs, Codes and Cryptography, 86 (2018), 239-250. doi: 10.1007/s10623-017-0360-6.

[29]

D. J. Kleitman, On an extremal property of antichains in partial orders. the LYM property and some of its implications and applications, in Combinatorics (eds. H. M. and van Lint J.H.), vol. 16 of NATO Advanced Study Institutes Series (Series C - Mathematical and Physical Sciences), Springer, Dordrecht, 1975,277-290. doi: 10.1007/978-94-010-1826-5_14.

[30]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Transactions on Information Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

[31]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, Springer, 5393 (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.

[32]

B. Segre, Teoria di galois, fibrazioni proiettive e geometrie non desarguesiane, Annali di Matematica Pura ed Applicata, 64 (1964), 1-76. doi: 10.1007/BF02410047.

[33]

D. SilvaF. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Transactions on Information Theory, 54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.

[34]

R. W. Yeung and N. Cai, Network error correction, Ⅰ: Basic concepts and upper bounds, Communications in Information & Systems, 6 (2006), 19-35. doi: 10.4310/CIS.2006.v6.n1.a2.

show all references

References:
[1]

R. Ahlswede and H. Aydinian, On error control codes for random network coding, in Network Coding, Theory, and Applications, 2009. NetCod'09. Workshop on, IEEE, 2009, 68-73. doi: 10.1109/NETCOD.2009.5191396.

[2]

C. BachocA. Passuello and F. Vallentin, Bounds for projective codes from semidefinite programming, Advances in Mathematics of Communications, 7 (2013), 127-145. doi: 10.3934/amc.2013.7.127.

[3]

M. BraunP. R. Östergård and A. Wassermann, New lower bounds for binary constant-dimension subspace codes, Experimental Mathematics, 27 (2018), 179-183. doi: 10.1080/10586458.2016.1239145.

[4]

M. Braun and J. Reichelt, $ q $-analogs of packing designs, Journal of Combinatorial Designs, 22 (2014), 306-321. doi: 10.1002/jcd.21376.

[5]

N. Cai and R. W. Yeung, Network error correction, Ⅱ: Lower bounds, Communications in Information & Systems, 6 (2006), 37-54. doi: 10.4310/CIS.2006.v6.n1.a3.

[6]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, Journal of Combinatorial Theory, Series A, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.

[7]

P. Dembowski, Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44 Springer-Verlag, Berlin-New York, 1968.

[8]

U. Dempwolff and A. Reifart, The classification of the translation planes of order 16, I, Geometriae Dedicata, 15 (1983), 137-153. doi: 10.1007/BF00147760.

[9]

T. EtzionE. GorlaA. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Transactions on Information Theory, 62 (2016), 1616-1630. doi: 10.1109/TIT.2016.2522971.

[10]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Transactions on Information Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.

[11]

T. Etzion and N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Transactions on Information Theory, 59 (2013), 1004-1017. doi: 10.1109/TIT.2012.2220119.

[12]

T. Etzion and L. Storme, Galois geometries and coding theory, Designs, Codes and Cryptography, 78 (2016), 311-350. doi: 10.1007/s10623-015-0156-5.

[13]

T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Advances in Mathematics of Communications, 3 (2009), 363-383. doi: 10.3934/amc.2009.3.363.

[14]

T. Feulner, Eine kanonische Form zur Darstellung äquivalenter Codes: Computergestützte Berechnung und ihre Anwendung in der Codierungstheorie, Kryptographie und Geometrie, PhD thesis, University of Bayreuth, Bayreuth, 2014.

[15]

E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.

[16]

M. Greferath, M. O. Pavčević, N. Silberstein and M. Á. Vázquez-Castro, Network Coding and Subspace Designs, Springer, 2018. doi: 10.1007/978-3-319-70293-3.

[17]

M. Hall JrJ. D. Swift and R. J. Walker, Uniqueness of the projective plane of order eight, Mathematical Tables and Other Aids to Computation, 10 (1956), 186-194. doi: 10.2307/2001913.

[18]

D. Heinlein, T. Honold, M. Kiermaier, S. Kurz and A. Wassermann, Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6 to appear in Designs, Codes and Cryptography, https://link.springer.com/article/10.1007%2Fs10623-018-0544-8. doi: 10.1007/s10623-018-0544-8.

[19]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864.

[20]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, A subspace code of size $333 $ in the setting of a binary $ q $-analog of the Fano plane, preprint, arXiv: 1708.06224.

[21]

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, World Scientific, 2016,141-175.

[22]

T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4, in Topics in Finite Fields, vol. 632 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2015,157-176. doi: 10.1090/conm/632/12627.

[23]

T. Honold, M. Kiermaier and S. Kurz, Classification of large partial plane spreads in $ {P}{G}(6, 2) $ and related combinatorial objects, to appear in Journal of Geometry. arXiv: 1606.07655.

[24]

T. HonoldM. Kiermaier and S. Kurz, Constructions and bounds for mixed-dimension subspace codes, Advances in Mathematics of Communications, 10 (2016), 649-682. doi: 10.3934/amc.2016033.

[25]

T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, Springer, 2018,131-170.

[26]

N. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes, CRC Press, 2007. doi: 10.1201/9781420011142.

[27]

M. Kiermaier and S. Kurz, An improvement of the Johnson bound for subspace codes, preprint, arXiv: 1707.00650.

[28]

M. KiermaierS. Kurz and A. Wassermann, The order of the automorphism group of a binary $ q $-analog of the Fano plane is at most two, Designs, Codes and Cryptography, 86 (2018), 239-250. doi: 10.1007/s10623-017-0360-6.

[29]

D. J. Kleitman, On an extremal property of antichains in partial orders. the LYM property and some of its implications and applications, in Combinatorics (eds. H. M. and van Lint J.H.), vol. 16 of NATO Advanced Study Institutes Series (Series C - Mathematical and Physical Sciences), Springer, Dordrecht, 1975,277-290. doi: 10.1007/978-94-010-1826-5_14.

[30]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Transactions on Information Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

[31]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, Springer, 5393 (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.

[32]

B. Segre, Teoria di galois, fibrazioni proiettive e geometrie non desarguesiane, Annali di Matematica Pura ed Applicata, 64 (1964), 1-76. doi: 10.1007/BF02410047.

[33]

D. SilvaF. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Transactions on Information Theory, 54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.

[34]

R. W. Yeung and N. Cai, Network error correction, Ⅰ: Basic concepts and upper bounds, Communications in Information & Systems, 6 (2006), 19-35. doi: 10.4310/CIS.2006.v6.n1.a2.

Table 1.  $\text{A}_2(v, d)\label{tbl:smax2}$ and isomorphism types of optimal codes for $v\leq 8$
$ v\backslash d $12345678
12(1)
25(1)3(1)
316(1)8(2)2(2)
467(1)37(1)5(4)5(1)
5374(1)187(2)18(48217)9(14)2(3)
62825(1)1521(1)108-11777(5)9(5)9(1)
729212(1)14606(2)614-776334-40734(39)17(1856)2(4)
8417199(1)222379(2)5687-92684803-6479263-326257(8)17(572)17(8)
$ v\backslash d $12345678
12(1)
25(1)3(1)
316(1)8(2)2(2)
467(1)37(1)5(4)5(1)
5374(1)187(2)18(48217)9(14)2(3)
62825(1)1521(1)108-11777(5)9(5)9(1)
729212(1)14606(2)614-776334-40734(39)17(1856)2(4)
8417199(1)222379(2)5687-92684803-6479263-326257(8)17(572)17(8)
Table 2.  Details for the ILP computations
IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$Orbits of phase 2 $\max z_8^{\operatorname{LP}}(\text{"31"})$ $\max z_8^{\operatorname{BLP}}(\text{"31"})$
116960272271.1856 $ 16^{2}, 240^{6}, 480^{47}, 960^{242} $263.0287799257
216384266.26086957267.4646 $ 96^{6}, 192^{91}, 384^{711} $206.04279728
3164270.83786676265.3281 $ 1^{13}, 2^{29}, 4^{2638} $257.20717665254
41648271.43451032262.082 $ 4^{3}, 12^{11}, 24^{59}, 48^{1104} $200.5850228
5162263.8132689259.8044 $ 1^{5}, 2^{59966} $206.39304042
61620267.53272206259.394 $ 5, 10^{9}, 20^{1843} $199.98690666
71764282.96047431259.1063 $ 16^{10}, 32^{145}, 64^{6293} $259.45364626257
81732268.0388109257.2408 $\le 255$ by a separate argumentation
IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$
9161263.82742528 $\le$ 25510161263.36961743 $\le$ 255
11161264.25957151 $\le$ 25412161263.85869815 $\le$ 254
13162263.07052878 $\le$ 254141612261.91860556 $\le$ 254
15164261.62648174 $\le$ 254161612261.31512837 $\le$ 254
17174261.11518721 $\le$ 25418161260.96388752 $\le$ 254
19161260.82432878 $\le$ 25420162260.65762276 $\le$ 254
21164260.43036283 $\le$ 25422162260.19475349 $\le$ 254
23161260.08583792 $\le$ 25424161260.04857193 $\le$ 254
25161259.75041996 $\le$ 25426162259.55230081 $\le$ 254
27162259.46335297 $\le$ 254281612259.11945025 $\le$ 254
29161258.89395938 $\le$ 254301724258.75142045 $\le$ 254
31168258.35689437 $\le$ 25432161257.81420526 $\le$ 254
33162257.75126819 $\le$ 25434164257.63965018 $\le$ 254
35161257.57663803 $\le$ 25436161257.2820438 $\le$ 254
37164257.01931801 $\le$ 2543817128257 $\le$ 254
391612256.83887168 $\le$ 254401612256.31380897 $\le$ 254
41166256.22093781 $\le$ 254421612256.10154389 $\le$ 254
43161255.87957119
IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$Orbits of phase 2 $\max z_8^{\operatorname{LP}}(\text{"31"})$ $\max z_8^{\operatorname{BLP}}(\text{"31"})$
116960272271.1856 $ 16^{2}, 240^{6}, 480^{47}, 960^{242} $263.0287799257
216384266.26086957267.4646 $ 96^{6}, 192^{91}, 384^{711} $206.04279728
3164270.83786676265.3281 $ 1^{13}, 2^{29}, 4^{2638} $257.20717665254
41648271.43451032262.082 $ 4^{3}, 12^{11}, 24^{59}, 48^{1104} $200.5850228
5162263.8132689259.8044 $ 1^{5}, 2^{59966} $206.39304042
61620267.53272206259.394 $ 5, 10^{9}, 20^{1843} $199.98690666
71764282.96047431259.1063 $ 16^{10}, 32^{145}, 64^{6293} $259.45364626257
81732268.0388109257.2408 $\le 255$ by a separate argumentation
IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$
9161263.82742528 $\le$ 25510161263.36961743 $\le$ 255
11161264.25957151 $\le$ 25412161263.85869815 $\le$ 254
13162263.07052878 $\le$ 254141612261.91860556 $\le$ 254
15164261.62648174 $\le$ 254161612261.31512837 $\le$ 254
17174261.11518721 $\le$ 25418161260.96388752 $\le$ 254
19161260.82432878 $\le$ 25420162260.65762276 $\le$ 254
21164260.43036283 $\le$ 25422162260.19475349 $\le$ 254
23161260.08583792 $\le$ 25424161260.04857193 $\le$ 254
25161259.75041996 $\le$ 25426162259.55230081 $\le$ 254
27162259.46335297 $\le$ 254281612259.11945025 $\le$ 254
29161258.89395938 $\le$ 254301724258.75142045 $\le$ 254
31168258.35689437 $\le$ 25432161257.81420526 $\le$ 254
33162257.75126819 $\le$ 25434164257.63965018 $\le$ 254
35161257.57663803 $\le$ 25436161257.2820438 $\le$ 254
37164257.01931801 $\le$ 2543817128257 $\le$ 254
391612256.83887168 $\le$ 254401612256.31380897 $\le$ 254
41166256.22093781 $\le$ 254421612256.10154389 $\le$ 254
43161255.87957119
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