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August  2016, 10(3): 525-540. doi: 10.3934/amc.2016023

Construction of subspace codes through linkage

Heide Gluesing-Luerssen 1, and  Carolyn Troha 2,

1. 

University of Kentucky, Department of Mathematics, Lexington, KY 40506-0027
2. 

Department of Mathematics, Viterbo University, La Crosse, WI, United States

Received  May 2015 Revised  September 2015 Published  August 2016

A construction is discussed that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting linkage code is as good as the minimum subspace distance of the constituent codes. As a special application, the construction of the best known partial spreads is reproduced. Finally, for a special case of linkage, a decoding algorithm is presented which amounts to decoding with respect to the smaller constituent codes and which can be parallelized.
Keywords: constant dimension subspace codes, partial spreads., Random network coding.
Mathematics Subject Classification: Primary: 11T71, 94B60; Secondary: 51E2.
Citation: Heide Gluesing-Luerssen, Carolyn Troha. Construction of subspace codes through linkage. Advances in Mathematics of Communications, 2016, 10 (3) : 525-540. doi: 10.3934/amc.2016023
References:
[1]

A. Beutelspacher, Partial spreads in finite projective spaces and partial designs, Math. Zeit., 145 (1975), 211-229.

[2]

M. Braun, T. Etzion, P. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems, Forum Math. PI, to appear.

[3]

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

[4]

T. Bu, Partitions of a vector space, Discr. Math., 31 (1980), 79-83. doi: 10.1016/0012-365X(80)90174-0.

[5]

J. de la Cruz, M. Kiermaier, A. Wassermann and W. Willems, Algebraic structures of MRD Codes, Adv. Math. Commun., 10 (2016), 499-510. doi: 10.3934/amc.2016021.

[6]

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.

[7]

D. A. Drake and J. W. Freeman, Partial $t$-spreads and group constructible $(s,r,\mu)$-nets, J. Geom., 13 (1979), 210-216. doi: 10.1007/BF01919756.

[8]

S. El-Zanati, H. Jordon, G. Seelinger, P. Sissokho and L. Spence, The maximum size of a partial $3$-spread in a finite vector space over $\mathbb F_2$ Des. Codes Cryptogr., 54 (2010), 101-107. doi: 10.1007/s10623-009-9311-1.

[9]

A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding, in Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, 2010, 1811-1814.

[10]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, IT-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 Trans. Inform. Theory, IT-59 (2013), 1004-1017. doi: 10.1109/TIT.2012.2220119.

[12]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, IT-57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232.

[13]

E. M. Gabidulin, Theory of codes with maximal rank distance, Probl. Inf. Transm., 21 (1985), 1-12.

[14]

E. M. Gabidulin and M. Bossert, Algebraic codes for network coding, Probl. Inf. Trans. (Engl. Transl.), 45 (2009), 343-356. doi: 10.1134/S003294600904005X.

[15]

E. M. Gabidulin and N. I. Pilipchuk, Rank subcodes in multicomponent network coding, Probl. Inf. Trans. (Engl. Transl.), 49 (2013), 40-53. doi: 10.1134/S0032946013010043.

[16]

E. M. Gabidulin, N. I. Pilipchuk and M. Bossert, Decoding of random network codes, Probl. Inf. Trans. (Engl. Transl.), 46 (2010), 300-320. doi: 10.1134/S0032946010040034.

[17]

H. Gluesing-Luerssen, K. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197. doi: 10.3934/amc.2015.9.177.

[18]

E. Gorla, F. Manganiello and J. Rosenthal, An algebraic approach for decoding spread codes, Adv. Math. Commun., 6 (2012), 443-466. doi: 10.3934/amc.2012.6.443.

[19]

E. Gorla and A. Ravagnani, Partial spreads in random network coding, Finite Fields Appl., 26 (2014), 104-115. doi: 10.1016/j.ffa.2013.11.007.

[20]

B. S. Hernandez and V. P. Sison, Grassmannian codes as lifts of matrix codes derived as images of linear block codes over finite fields, Global J. Pure Appl. Math., 12 (2016), 1801-1820.

[21]

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

[22]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in Proc. 12th IMA Conf. Crypt. Coding, Cirencester, 2009, 1-21. doi: 10.1007/978-3-642-10868-6_1.

[23]

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

[24]

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

[25]

J. Rosenthal and A.-L. Trautmann, A complete characterization of irreducible cyclic orbit codes and their Plücker embedding, Des. Codes Cryptogr., 66 (2013), 275-289. doi: 10.1007/s10623-012-9691-5.

[26]

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

[27]

D. Silva and F. R. Kschischang, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, IT-55 (2009), 5479-5490. doi: 10.1109/TIT.2009.2032817.

[28]

D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, IT-54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.

[29]

V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, IT-56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.

[30]

A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inform. Theory, IT-59 (2013), 7386-7404. doi: 10.1109/TIT.2013.2274266.

[31]

A.-L. Trautmann and J. Rosenthal, New improvements on the Echelon Ferrers construction, in Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, 2010, 405-408.

show all references

References:
[1]

A. Beutelspacher, Partial spreads in finite projective spaces and partial designs, Math. Zeit., 145 (1975), 211-229.

[2]

M. Braun, T. Etzion, P. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems, Forum Math. PI, to appear.

[3]

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

[4]

T. Bu, Partitions of a vector space, Discr. Math., 31 (1980), 79-83. doi: 10.1016/0012-365X(80)90174-0.

[5]

J. de la Cruz, M. Kiermaier, A. Wassermann and W. Willems, Algebraic structures of MRD Codes, Adv. Math. Commun., 10 (2016), 499-510. doi: 10.3934/amc.2016021.

[6]

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.

[7]

D. A. Drake and J. W. Freeman, Partial $t$-spreads and group constructible $(s,r,\mu)$-nets, J. Geom., 13 (1979), 210-216. doi: 10.1007/BF01919756.

[8]

S. El-Zanati, H. Jordon, G. Seelinger, P. Sissokho and L. Spence, The maximum size of a partial $3$-spread in a finite vector space over $\mathbb F_2$ Des. Codes Cryptogr., 54 (2010), 101-107. doi: 10.1007/s10623-009-9311-1.

[9]

A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding, in Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, 2010, 1811-1814.

[10]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, IT-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 Trans. Inform. Theory, IT-59 (2013), 1004-1017. doi: 10.1109/TIT.2012.2220119.

[12]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, IT-57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232.

[13]

E. M. Gabidulin, Theory of codes with maximal rank distance, Probl. Inf. Transm., 21 (1985), 1-12.

[14]

E. M. Gabidulin and M. Bossert, Algebraic codes for network coding, Probl. Inf. Trans. (Engl. Transl.), 45 (2009), 343-356. doi: 10.1134/S003294600904005X.

[15]

E. M. Gabidulin and N. I. Pilipchuk, Rank subcodes in multicomponent network coding, Probl. Inf. Trans. (Engl. Transl.), 49 (2013), 40-53. doi: 10.1134/S0032946013010043.

[16]

E. M. Gabidulin, N. I. Pilipchuk and M. Bossert, Decoding of random network codes, Probl. Inf. Trans. (Engl. Transl.), 46 (2010), 300-320. doi: 10.1134/S0032946010040034.

[17]

H. Gluesing-Luerssen, K. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197. doi: 10.3934/amc.2015.9.177.

[18]

E. Gorla, F. Manganiello and J. Rosenthal, An algebraic approach for decoding spread codes, Adv. Math. Commun., 6 (2012), 443-466. doi: 10.3934/amc.2012.6.443.

[19]

E. Gorla and A. Ravagnani, Partial spreads in random network coding, Finite Fields Appl., 26 (2014), 104-115. doi: 10.1016/j.ffa.2013.11.007.

[20]

B. S. Hernandez and V. P. Sison, Grassmannian codes as lifts of matrix codes derived as images of linear block codes over finite fields, Global J. Pure Appl. Math., 12 (2016), 1801-1820.

[21]

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

[22]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in Proc. 12th IMA Conf. Crypt. Coding, Cirencester, 2009, 1-21. doi: 10.1007/978-3-642-10868-6_1.

[23]

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

[24]

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

[25]

J. Rosenthal and A.-L. Trautmann, A complete characterization of irreducible cyclic orbit codes and their Plücker embedding, Des. Codes Cryptogr., 66 (2013), 275-289. doi: 10.1007/s10623-012-9691-5.

[26]

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

[27]

D. Silva and F. R. Kschischang, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, IT-55 (2009), 5479-5490. doi: 10.1109/TIT.2009.2032817.

[28]

D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, IT-54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.

[29]

V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, IT-56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.

[30]

A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inform. Theory, IT-59 (2013), 7386-7404. doi: 10.1109/TIT.2013.2274266.

[31]

A.-L. Trautmann and J. Rosenthal, New improvements on the Echelon Ferrers construction, in Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, 2010, 405-408.

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