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doi: 10.3934/amc.2021007
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Combining subspace codes

Antonio Cossidente 1, , Sascha Kurz 2, , Giuseppe Marino 3, and  Francesco Pavese 4,

1. 

Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Contrada Macchia Romana, 85100, Potenza, Italy
2. 

Mathematisches Institut, Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany
3. 

Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Complesso Universitario di Monte Sant'Angelo, Cupa Nuova Cintia 21, 80126, Napoli, Italy
4. 

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy

Received  November 2020 Revised  February 2021 Early access April 2021

Fund Project: The work of the first, third and fourth author was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA{ INdAM)

In the context of constant-dimension subspace codes, an important problem is to determine the largest possible size $ A_q(n, d; k) $ of codes whose codewords are $ k $-subspaces of $ {\mathbb F}_q^n $ with minimum subspace distance $ d $. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant-dimension subspace codes for many parameters, including $ A_q(10, 4; 5) $, $ A_q(12, 4; 4) $, $ A_q(12, 6, 6) $ and $ A_q(16, 4; 4) $.

Keywords: Constant-dimension subspace code, finite projective geometry, network coding.
Mathematics Subject Classification: Primary: 51E20; Secondary: 05B25, 94B65.
Citation: Antonio Cossidente, Sascha Kurz, Giuseppe Marino, Francesco Pavese. Combining subspace codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021007
References:
[1]

J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z., 60 (1954), 156-186.  doi: 10.1007/BF01187370.

[2]

A. Beutelspacher, On parallelisms in finite projective spaces, Geometriae Dedicata, 3 (1974), 35-40.  doi: 10.1007/BF00181359.

[3]

M. Braun, P. R. J. Östergård and A. Wassermann, New lower bounds for binary constant-dimension subspace codes, Exp. Math., 27 (2018), 179-183.  doi: 10.1080/10586458.2016.1239145.

[4]

H. Chen, X. He, J. Weng and L. Xu, New constructions of subspace codes using subsets of MRD codes in several blocks, IEEE Trans. Inform. Theory, 66 (2020), 5317-5321.  doi: 10.1109/TIT.2020.2975776.

[5]

A. Cossidente, G. Marino and F. Pavese, Subspace code constructions, Ric. di Mat., (2020). to appear. doi: 10.1007/s11587-020-00521-9.

[6]

A. Cossidente and F. Pavese, On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.  doi: 10.1007/s10623-014-0018-6.

[7]

A. Cossidente and F. Pavese, Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.  doi: 10.1007/s10623-015-0166-3.

[8]

A. Cossidente and F. Pavese, Subspace codes in ${\rm{PG(2N - 1, Q)}}$, Combinatorica, 37 (2017), 1073-1095.  doi: 10.1007/s00493-016-3354-5.

[9]

A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 107-129.

[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]

R. H. F. Denniston, Some packings of projective spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 52 (1972), 36-40. 

[12]

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

[13]

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

[14]

P. Frankl and V. Rödl, Near perfect coverings in graphs and hypergraphs, European J. Combin., 6 (1985), 317-326.  doi: 10.1016/S0195-6698(85)80045-7.

[15]

M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.  doi: 10.1109/TIT.2010.2048447.

[16]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.  doi: 10.3934/amc.2016023.

[17]

X. He, Construction of constant dimension code from two parallel versions of linkage construction, IEEE Communi. Lett., 24 (2020), 2392-2395. 

[18]

X. He and Y. Chen, Construction of constant dimension codes from several parallel lifted MRD code, preprint, arXiv: 1911.00154.

[19]

D. Heinlein, New LMRD code bounds for constant dimension codes and improved constructions, IEEE Trans. Inform. Theory, 65 (2019), 4822-4830.  doi: 10.1109/TIT.2019.2905002.

[20]

D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Trans. Inform. Theory, 67 (2020), 705-715. 

[21]

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 $6$, Des. Codes Cryptogr., 87 (2019), 375-391.  doi: 10.1007/s10623-018-0544-8.

[22]

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

[23]

D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, in Coding Theory and Applications : 5th International Castle Meeting, ICMCTA 2017, Vihula, Estonia, August 28-31, 2017, Proceedings, vol. 10495 of Lecture Notes in Computer Science, Springer International Publishing, Cham, (2017), 163-191. doi: 10.1007/978-3-319-66278-7_15.

[24]

D. Heinlein and S. Kurz, Coset construction for subspace codes, IEEE Trans. Inform. Theory, 63 (2017), 7651-7660.  doi: 10.1109/TIT.2017.2753822.

[25]

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.

[26]

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

[27]

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

[28]

A.-L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 25-42.

[29]

M. Kiermaier and S. Kurz, On the lengths of divisible codes, IEEE Trans. Inform. Theory, 66 (2020), 4051-4060.  doi: 10.1109/TIT.2020.2968832.

[30]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. 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, (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.

[32]

S. Kurz, Packing vector spaces into vector spaces, Australas. J. Combin., 68 (2017), 122-130. 

[33]

S. Kurz, A note on the linkage construction for constant dimension codes, preprint, arXiv: 1906.09780.

[34]

S. Kurz, Subspaces intersecting in at most a point, Des. Codes Cryptogr., 88 (2020), 595-599.  doi: 10.1007/s10623-019-00699-6.

[35]

S. Liu, Y. Chang and T. Feng, Parallel multilevel construction for constant dimension codes, IEEE Trans. Inform. Theory, 66 (2020), 6884-6897.  doi: 10.1109/TIT.2020.3004315.

[36]

E. L. Năstase and P. A. Sissokho, The maximum size of a partial spread in a finite projective space, J. Combin. Theory Ser. A, 152 (2017), 353-362.  doi: 10.1016/j.jcta.2017.06.012.

[37]

B. Segre, Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.  doi: 10.1007/BF02410047.

[38]

J. Sheekey, MRD codes: Constructions and connections, in Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2019. doi: 10.1515/9783110642094-013.

[39]

N. Silberstein and A.-L. Trautmann, Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.  doi: 10.1109/TIT.2015.2435743.

[40]

D. Silva, F. R. Kschischang and R. Köetter, 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.

[41]

H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: Bounds and constructions, IEEE Trans. Inform. Theory, 49 (2003), 866-872.  doi: 10.1109/TIT.2003.809567.

[42]

S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.  doi: 10.1007/s10623-008-9221-7.

[43]

L. Xu and H. Chen, New constant-dimension subspace codes from maximum rank distance codes, IEEE Trans. Inform. Theory, 64 (2018), 6315-6319.  doi: 10.1109/TIT.2018.2839596.

show all references

References:
[1]

J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z., 60 (1954), 156-186.  doi: 10.1007/BF01187370.

[2]

A. Beutelspacher, On parallelisms in finite projective spaces, Geometriae Dedicata, 3 (1974), 35-40.  doi: 10.1007/BF00181359.

[3]

M. Braun, P. R. J. Östergård and A. Wassermann, New lower bounds for binary constant-dimension subspace codes, Exp. Math., 27 (2018), 179-183.  doi: 10.1080/10586458.2016.1239145.

[4]

H. Chen, X. He, J. Weng and L. Xu, New constructions of subspace codes using subsets of MRD codes in several blocks, IEEE Trans. Inform. Theory, 66 (2020), 5317-5321.  doi: 10.1109/TIT.2020.2975776.

[5]

A. Cossidente, G. Marino and F. Pavese, Subspace code constructions, Ric. di Mat., (2020). to appear. doi: 10.1007/s11587-020-00521-9.

[6]

A. Cossidente and F. Pavese, On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.  doi: 10.1007/s10623-014-0018-6.

[7]

A. Cossidente and F. Pavese, Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.  doi: 10.1007/s10623-015-0166-3.

[8]

A. Cossidente and F. Pavese, Subspace codes in ${\rm{PG(2N - 1, Q)}}$, Combinatorica, 37 (2017), 1073-1095.  doi: 10.1007/s00493-016-3354-5.

[9]

A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 107-129.

[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]

R. H. F. Denniston, Some packings of projective spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 52 (1972), 36-40. 

[12]

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

[13]

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

[14]

P. Frankl and V. Rödl, Near perfect coverings in graphs and hypergraphs, European J. Combin., 6 (1985), 317-326.  doi: 10.1016/S0195-6698(85)80045-7.

[15]

M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.  doi: 10.1109/TIT.2010.2048447.

[16]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.  doi: 10.3934/amc.2016023.

[17]

X. He, Construction of constant dimension code from two parallel versions of linkage construction, IEEE Communi. Lett., 24 (2020), 2392-2395. 

[18]

X. He and Y. Chen, Construction of constant dimension codes from several parallel lifted MRD code, preprint, arXiv: 1911.00154.

[19]

D. Heinlein, New LMRD code bounds for constant dimension codes and improved constructions, IEEE Trans. Inform. Theory, 65 (2019), 4822-4830.  doi: 10.1109/TIT.2019.2905002.

[20]

D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Trans. Inform. Theory, 67 (2020), 705-715. 

[21]

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 $6$, Des. Codes Cryptogr., 87 (2019), 375-391.  doi: 10.1007/s10623-018-0544-8.

[22]

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

[23]

D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, in Coding Theory and Applications : 5th International Castle Meeting, ICMCTA 2017, Vihula, Estonia, August 28-31, 2017, Proceedings, vol. 10495 of Lecture Notes in Computer Science, Springer International Publishing, Cham, (2017), 163-191. doi: 10.1007/978-3-319-66278-7_15.

[24]

D. Heinlein and S. Kurz, Coset construction for subspace codes, IEEE Trans. Inform. Theory, 63 (2017), 7651-7660.  doi: 10.1109/TIT.2017.2753822.

[25]

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.

[26]

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

[27]

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

[28]

A.-L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 25-42.

[29]

M. Kiermaier and S. Kurz, On the lengths of divisible codes, IEEE Trans. Inform. Theory, 66 (2020), 4051-4060.  doi: 10.1109/TIT.2020.2968832.

[30]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. 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, (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.

[32]

S. Kurz, Packing vector spaces into vector spaces, Australas. J. Combin., 68 (2017), 122-130. 

[33]

S. Kurz, A note on the linkage construction for constant dimension codes, preprint, arXiv: 1906.09780.

[34]

S. Kurz, Subspaces intersecting in at most a point, Des. Codes Cryptogr., 88 (2020), 595-599.  doi: 10.1007/s10623-019-00699-6.

[35]

S. Liu, Y. Chang and T. Feng, Parallel multilevel construction for constant dimension codes, IEEE Trans. Inform. Theory, 66 (2020), 6884-6897.  doi: 10.1109/TIT.2020.3004315.

[36]

E. L. Năstase and P. A. Sissokho, The maximum size of a partial spread in a finite projective space, J. Combin. Theory Ser. A, 152 (2017), 353-362.  doi: 10.1016/j.jcta.2017.06.012.

[37]

B. Segre, Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.  doi: 10.1007/BF02410047.

[38]

J. Sheekey, MRD codes: Constructions and connections, in Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2019. doi: 10.1515/9783110642094-013.

[39]

N. Silberstein and A.-L. Trautmann, Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.  doi: 10.1109/TIT.2015.2435743.

[40]

D. Silva, F. R. Kschischang and R. Köetter, 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.

[41]

H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: Bounds and constructions, IEEE Trans. Inform. Theory, 49 (2003), 866-872.  doi: 10.1109/TIT.2003.809567.

[42]

S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.  doi: 10.1007/s10623-008-9221-7.

[43]

L. Xu and H. Chen, New constant-dimension subspace codes from maximum rank distance codes, IEEE Trans. Inform. Theory, 64 (2018), 6315-6319.  doi: 10.1109/TIT.2018.2839596.

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