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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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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