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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. Google Scholar

[2]

M. Braun, T. Etzion, P. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems,, Forum Math. PI, (). Google Scholar

[3]

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

[4]

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

[5]

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

[6]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory,, J. Combin. Theory Ser. A, 25 (1978), 226. doi: 10.1016/0097-3165(78)90015-8. Google Scholar

[7]

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

[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. doi: 10.1007/s10623-009-9311-1. Google Scholar

[9]

A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding,, in Proc. 19th Int. Symp. Math. Theory Netw. Syst., (2010), 1811. Google Scholar

[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. doi: 10.1109/TIT.2009.2021376. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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. Google Scholar

[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. doi: 10.1090/conm/632/12627. Google Scholar

[22]

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

[23]

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

[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, (2008), 31. doi: 10.1007/978-3-540-89994-5_4. Google Scholar

[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. doi: 10.1007/s10623-012-9691-5. Google Scholar

[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. doi: 10.1109/TIT.2015.2435743. Google Scholar

[27]

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

[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. doi: 10.1109/TIT.2008.928291. Google Scholar

[29]

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

[30]

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

[31]

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

show all references

References:
[1]

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

[2]

M. Braun, T. Etzion, P. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems,, Forum Math. PI, (). Google Scholar

[3]

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

[4]

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

[5]

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

[6]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory,, J. Combin. Theory Ser. A, 25 (1978), 226. doi: 10.1016/0097-3165(78)90015-8. Google Scholar

[7]

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

[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. doi: 10.1007/s10623-009-9311-1. Google Scholar

[9]

A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding,, in Proc. 19th Int. Symp. Math. Theory Netw. Syst., (2010), 1811. Google Scholar

[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. doi: 10.1109/TIT.2009.2021376. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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. Google Scholar

[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. doi: 10.1090/conm/632/12627. Google Scholar

[22]

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

[23]

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

[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, (2008), 31. doi: 10.1007/978-3-540-89994-5_4. Google Scholar

[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. doi: 10.1007/s10623-012-9691-5. Google Scholar

[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. doi: 10.1109/TIT.2015.2435743. Google Scholar

[27]

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

[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. doi: 10.1109/TIT.2008.928291. Google Scholar

[29]

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

[30]

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

[31]

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

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