# American Institute of Mathematical Sciences

November 2017, 11(4): 705-713. doi: 10.3934/amc.2017051

## Constant dimension codes from Riemann-Roch spaces

 1 Department of Mathematics and Computer Science, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy 2 Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (Trento), Italy

Received  February 2016 Revised  June 2017 Published  November 2017

Fund Project: The research of D. Bartoli and M. Giulietti was supported by Ministry for Education, University and Research of Italy (MIUR) (Project "Geometrie di Galois e strutture di incidenza") and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA -INdAM)

Some families of constant dimension codes arising from Riemann-Roch spaces associated with particular divisors of a curve $\mathcal{X}$ are constructed. These families are generalizations of the one constructed by Hansen.

Citation: Daniele Bartoli, Matteo Bonini, Massimo Giulietti. Constant dimension codes from Riemann-Roch spaces. Advances in Mathematics of Communications, 2017, 11 (4) : 705-713. doi: 10.3934/amc.2017051
##### References:
 [1] D. Bartoli and F. Pavese, A note on Equidistant subspaces codes, Discrete Appl. Math., 198 (2016), 291-296. doi: 10.1016/j.dam.2015.06.017. [2] A. Cossidente and F. Pavese, On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531. doi: 10.1007/s10623-014-0018-6. [3] 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. [4] 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. [5] T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232. [6] 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. [7] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd edition, 1994. [8] J. P. Hansen, Riemann-Roch theorem and linear network codes, Int. J. Math. Comput. Sci., 10 (2015), 1-11. [9] T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4, in Topics in Finite Fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott Editors, Contemporary Mathematics, 632 (2015), 157-176. [10] A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, Cryptography and coding, in Lecture Notes in Compututer Science, Springer, Berlin, 5921 (2009), 1-21. [11] R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. on Inform. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449. [12] D. Silva, F. R. Kschischang and R. Koetter, 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. [13] A. L. Trautmann and J. Rosenthal, New improvements on the echelon-ferrers construction, in Proc. Int. Symp. on Math. Theory of Networks and Systems, 2010,405-408

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##### References:
 [1] D. Bartoli and F. Pavese, A note on Equidistant subspaces codes, Discrete Appl. Math., 198 (2016), 291-296. doi: 10.1016/j.dam.2015.06.017. [2] A. Cossidente and F. Pavese, On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531. doi: 10.1007/s10623-014-0018-6. [3] 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. [4] 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. [5] T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232. [6] 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. [7] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd edition, 1994. [8] J. P. Hansen, Riemann-Roch theorem and linear network codes, Int. J. Math. Comput. Sci., 10 (2015), 1-11. [9] T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4, in Topics in Finite Fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott Editors, Contemporary Mathematics, 632 (2015), 157-176. [10] A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, Cryptography and coding, in Lecture Notes in Compututer Science, Springer, Berlin, 5921 (2009), 1-21. [11] R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. on Inform. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449. [12] D. Silva, F. R. Kschischang and R. Koetter, 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. [13] A. L. Trautmann and J. Rosenthal, New improvements on the echelon-ferrers construction, in Proc. Int. Symp. on Math. Theory of Networks and Systems, 2010,405-408
Normalized weight, rate, and normalized minimal distance, $s>1$
 Normalized weight Rate Normalized minimum distance $\mathcal{H}_{k,s}$ $\frac{ks+1-g}{nk+1-g}$ $\frac{\log_q \binom{n}{s}}{(nk+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$ $\mathcal{A}_{k,s}$ $\frac{ks+1-g}{nks+1-g}$ $\frac{\log_q \binom{n+s-1}{s}}{(nks+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$ $\mathcal{B}_{k,s,w}$ $\frac{ks+1-g}{nkw+1-g}$ $\frac{\log_q U_{n,s,0,w}}{(nkw+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$ $\mathcal{C}_{k,s,w}$ $\frac{ks+1-g}{nkw+1-g}$ $\frac{\log_q U^{\prime}_{n,s,s-w(n-1),w}}{(nkw+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$
 Normalized weight Rate Normalized minimum distance $\mathcal{H}_{k,s}$ $\frac{ks+1-g}{nk+1-g}$ $\frac{\log_q \binom{n}{s}}{(nk+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$ $\mathcal{A}_{k,s}$ $\frac{ks+1-g}{nks+1-g}$ $\frac{\log_q \binom{n+s-1}{s}}{(nks+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$ $\mathcal{B}_{k,s,w}$ $\frac{ks+1-g}{nkw+1-g}$ $\frac{\log_q U_{n,s,0,w}}{(nkw+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$ $\mathcal{C}_{k,s,w}$ $\frac{ks+1-g}{nkw+1-g}$ $\frac{\log_q U^{\prime}_{n,s,s-w(n-1),w}}{(nkw+1-g)(ks+1-g)}$ $\frac{1}{s+\frac{1-g}{k}}$
Normalized weight, rate, and normalized minimal distance for $g=1$, $s>1$
 Normalized weight Rate Normalized minimum distance $\mathcal{H}_{k,s}$ $\frac{s}{n}$ $\frac{\log_q \binom{n}{s}}{nk^2s}$ $\frac{1}{s}$ $\mathcal{A}_{k,s}$ $\frac{1}{s}$ $\frac{\log_q \binom{n+s-1}{s}}{nk^2s^2}$ $\frac{1}{s}$ $\mathcal{B}_{k,s,w}$ $\frac{s}{nw}$ $\frac{\log_q U_{n,s,0,w}}{nk^2ws}$ $\frac{1}{s}$ $\mathcal{C}_{k,s,w}$ $\frac{s}{nw}$ $\frac{\log_q U^{\prime}_{n,s,s-w(n-1),w}}{nk^2ws}$ $\frac{1}{s}$
 Normalized weight Rate Normalized minimum distance $\mathcal{H}_{k,s}$ $\frac{s}{n}$ $\frac{\log_q \binom{n}{s}}{nk^2s}$ $\frac{1}{s}$ $\mathcal{A}_{k,s}$ $\frac{1}{s}$ $\frac{\log_q \binom{n+s-1}{s}}{nk^2s^2}$ $\frac{1}{s}$ $\mathcal{B}_{k,s,w}$ $\frac{s}{nw}$ $\frac{\log_q U_{n,s,0,w}}{nk^2ws}$ $\frac{1}{s}$ $\mathcal{C}_{k,s,w}$ $\frac{s}{nw}$ $\frac{\log_q U^{\prime}_{n,s,s-w(n-1),w}}{nk^2ws}$ $\frac{1}{s}$
Rates of $\mathcal{H}_{k,s}$, $\mathcal{A}_{k,s}$, $\mathcal{B}_{k,s,w}$, $\mathcal{C}_{k,s,w}$ for $q=16$, $8\leq n\leq 14$, $1\leq s <n$, $w=3$, $k=5$
 $(n,s)$ $\mathcal{H}_{k,s}$ $\mathcal{A}_{k,s}$ $\mathcal{B}_{k,s,w}$ $\mathcal{C}_{k,s,w}$ $(8,1)$ $0.003750$ $0.003750$ $0.001250$ $0.008732$ $(8,2)$ $0.003005$ $0.001616$ $0.001077$ $0.004286$ $(8,3)$ $0.002420$ $0.000959$ $0.000959$ $0.002802$ $(8,4)$ $0.001915$ $0.000654$ $0.000868$ $0.002058$ $(8,5)$ $0.001452$ $0.000481$ $0.000792$ $0.001611$ $(8,6)$ $0.001002$ $0.000373$ $0.000728$ $0.001311$ $(8,7)$ $0.000536$ $0.000300$ $0.000671$ $0.001095$ $(9,1)$ $0.003522$ $0.003522$ $0.001174$ $0.008931$ $(9,2)$ $0.002872$ $0.001526$ $0.001017$ $0.004394$ $(9,3)$ $0.002368$ $0.000909$ $0.000909$ $0.002880$ $(9,4)$ $0.001938$ $0.000622$ $0.000826$ $0.002121$ $(9,5)$ $0.001551$ $0.000459$ $0.000758$ $0.001665$ $(9,6)$ $0.001184$ $0.000357$ $0.000700$ $0.001360$ $(9,7)$ $0.000821$ $0.000287$ $0.000649$ $0.001141$ $(9,8)$ $0.000440$ $0.000237$ $0.000604$ $0.000976$ $(10,1)$ $0.003322$ $0.003322$ $0.001107$ $0.009093$ $(10,2)$ $0.002746$ $0.001445$ $0.000964$ $0.004482$ $(10,3)$ $0.002302$ $0.000865$ $0.000865$ $0.002943$ $(10,4)$ $0.001929$ $0.000593$ $0.000788$ $0.002173$ $(10,5)$ $0.001595$ $0.000439$ $0.000726$ $0.001710$ $(10,6)$ $0.001286$ $0.000341$ $0.000673$ $0.001400$ $(10,7)$ $0.000987$ $0.000275$ $0.000627$ $0.001178$ $(10,8)$ $0.000686$ $0.000228$ $0.000586$ $0.001011$ $(10,9)$ $0.000369$ $0.000192$ $0.000549$ $0.000880$ $(11,1)$ $0.003145$ $0.003145$ $0.001048$ $0.009229$ $(11,2)$ $0.002628$ $0.001374$ $0.000916$ $0.004555$ $(11,3)$ $0.002232$ $0.000824$ $0.000824$ $0.002996$ $(11,4)$ $0.001901$ $0.000566$ $0.000754$ $0.002215$ $(11,5)$ $0.001609$ $0.000420$ $0.000697$ $0.001746$ $(11,6)$ $0.001341$ $0.000327$ $0.000648$ $0.001433$ $(11,7)$ $0.001087$ $0.000264$ $0.000606$ $0.001209$ $(11,8)$ $0.000837$ $0.000219$ $0.000568$ $0.001040$ $(11,9)$ $0.000584$ $0.000185$ $0.000534$ $0.000908$ $(11,10)$ $0.000314$ $0.000159$ $0.000503$ $0.000802$ $(12,1)$ $0.002987$ $0.002987$ $0.000996$ $0.009343$ $(12,2)$ $0.002518$ $0.001309$ $0.000873$ $0.004617$ $(12,3)$ $0.002161$ $0.000788$ $0.000788$ $0.003040$ $(12,4)$ $0.001865$ $0.000542$ $0.000722$ $0.002251$ $(12,5)$ $0.001605$ $0.000403$ $0.000669$ $0.001778$ $(12,6)$ $0.001368$ $0.000315$ $0.000624$ $0.001461$ $(12,7)$ $0.001146$ $0.000254$ $0.000585$ $0.001234$ $(12,8)$ $0.000932$ $0.000211$ $0.000551$ $0.001064$ $(12,9)$ $0.000720$ $0.000179$ $0.000519$ $0.000931$ $(12,10)$ $0.000504$ $0.000154$ $0.000491$ $0.000824$ $(12,11)$ $0.000272$ $0.000134$ $0.000465$ $0.000736$ $(13,1)$ $0.002846$ $0.002846$ $0.000949$ $0.009440$ $(13,2)$ $0.002417$ $0.001251$ $0.000834$ $0.004670$ $(13,3)$ $0.002092$ $0.000755$ $0.000755$ $0.003079$ $(13,4)$ $0.001823$ $0.000521$ $0.000694$ $0.002282$ $(13,5)$ $0.001589$ $0.000388$ $0.000644$ $0.001804$ $(13,6)$ $0.001378$ $0.000303$ $0.000602$ $0.001485$ $(13,7)$ $0.001181$ $0.000245$ $0.000566$ $0.001256$ $(13,8)$ $0.000993$ $0.000204$ $0.000533$ $0.001085$ $(13,9)$ $0.000810$ $0.000173$ $0.000505$ $0.000950$ $(13,10)$ $0.000628$ $0.000148$ $0.000478$ $0.000843$ $(13,11)$ $0.000440$ $0.000129$ $0.000454$ $0.000755$ $(13,12)$ $0.000237$ $0.000114$ $0.000432$ $0.000681$ $(14,1)$ $0.002720$ $0.002720$ $0.000907$ $0.009486$ $(14,2)$ $0.002324$ $0.001199$ $0.000799$ $0.004705$ $(14,3)$ $0.002026$ $0.000725$ $0.000725$ $0.003102$ $(14,4)$ $0.001780$ $0.000501$ $0.000667$ $0.002307$ $(14,5)$ $0.001567$ $0.000373$ $0.000621$ $0.001827$ $(14,6)$ $0.001375$ $0.000292$ $0.000581$ $0.001511$ $(14,7)$ $0.001198$ $0.000237$ $0.000547$ $0.001252$ $(14,8)$ $0.001031$ $0.000197$ $0.000517$ $0.001106$ $(14,9)$ $0.000870$ $0.000167$ $0.000490$ $0.000974$ $(14,10)$ $0.000712$ $0.000144$ $0.000466$ $0.000865$ $(14,11)$ $0.000552$ $0.000125$ $0.000443$ $0.000768$ $(14,12)$ $0.000387$ $0.000111$ $0.000422$ $0.000699$ $(14,13)$ $0.000209$ $0.000099$ $0.000403$ $0.000635$
 $(n,s)$ $\mathcal{H}_{k,s}$ $\mathcal{A}_{k,s}$ $\mathcal{B}_{k,s,w}$ $\mathcal{C}_{k,s,w}$ $(8,1)$ $0.003750$ $0.003750$ $0.001250$ $0.008732$ $(8,2)$ $0.003005$ $0.001616$ $0.001077$ $0.004286$ $(8,3)$ $0.002420$ $0.000959$ $0.000959$ $0.002802$ $(8,4)$ $0.001915$ $0.000654$ $0.000868$ $0.002058$ $(8,5)$ $0.001452$ $0.000481$ $0.000792$ $0.001611$ $(8,6)$ $0.001002$ $0.000373$ $0.000728$ $0.001311$ $(8,7)$ $0.000536$ $0.000300$ $0.000671$ $0.001095$ $(9,1)$ $0.003522$ $0.003522$ $0.001174$ $0.008931$ $(9,2)$ $0.002872$ $0.001526$ $0.001017$ $0.004394$ $(9,3)$ $0.002368$ $0.000909$ $0.000909$ $0.002880$ $(9,4)$ $0.001938$ $0.000622$ $0.000826$ $0.002121$ $(9,5)$ $0.001551$ $0.000459$ $0.000758$ $0.001665$ $(9,6)$ $0.001184$ $0.000357$ $0.000700$ $0.001360$ $(9,7)$ $0.000821$ $0.000287$ $0.000649$ $0.001141$ $(9,8)$ $0.000440$ $0.000237$ $0.000604$ $0.000976$ $(10,1)$ $0.003322$ $0.003322$ $0.001107$ $0.009093$ $(10,2)$ $0.002746$ $0.001445$ $0.000964$ $0.004482$ $(10,3)$ $0.002302$ $0.000865$ $0.000865$ $0.002943$ $(10,4)$ $0.001929$ $0.000593$ $0.000788$ $0.002173$ $(10,5)$ $0.001595$ $0.000439$ $0.000726$ $0.001710$ $(10,6)$ $0.001286$ $0.000341$ $0.000673$ $0.001400$ $(10,7)$ $0.000987$ $0.000275$ $0.000627$ $0.001178$ $(10,8)$ $0.000686$ $0.000228$ $0.000586$ $0.001011$ $(10,9)$ $0.000369$ $0.000192$ $0.000549$ $0.000880$ $(11,1)$ $0.003145$ $0.003145$ $0.001048$ $0.009229$ $(11,2)$ $0.002628$ $0.001374$ $0.000916$ $0.004555$ $(11,3)$ $0.002232$ $0.000824$ $0.000824$ $0.002996$ $(11,4)$ $0.001901$ $0.000566$ $0.000754$ $0.002215$ $(11,5)$ $0.001609$ $0.000420$ $0.000697$ $0.001746$ $(11,6)$ $0.001341$ $0.000327$ $0.000648$ $0.001433$ $(11,7)$ $0.001087$ $0.000264$ $0.000606$ $0.001209$ $(11,8)$ $0.000837$ $0.000219$ $0.000568$ $0.001040$ $(11,9)$ $0.000584$ $0.000185$ $0.000534$ $0.000908$ $(11,10)$ $0.000314$ $0.000159$ $0.000503$ $0.000802$ $(12,1)$ $0.002987$ $0.002987$ $0.000996$ $0.009343$ $(12,2)$ $0.002518$ $0.001309$ $0.000873$ $0.004617$ $(12,3)$ $0.002161$ $0.000788$ $0.000788$ $0.003040$ $(12,4)$ $0.001865$ $0.000542$ $0.000722$ $0.002251$ $(12,5)$ $0.001605$ $0.000403$ $0.000669$ $0.001778$ $(12,6)$ $0.001368$ $0.000315$ $0.000624$ $0.001461$ $(12,7)$ $0.001146$ $0.000254$ $0.000585$ $0.001234$ $(12,8)$ $0.000932$ $0.000211$ $0.000551$ $0.001064$ $(12,9)$ $0.000720$ $0.000179$ $0.000519$ $0.000931$ $(12,10)$ $0.000504$ $0.000154$ $0.000491$ $0.000824$ $(12,11)$ $0.000272$ $0.000134$ $0.000465$ $0.000736$ $(13,1)$ $0.002846$ $0.002846$ $0.000949$ $0.009440$ $(13,2)$ $0.002417$ $0.001251$ $0.000834$ $0.004670$ $(13,3)$ $0.002092$ $0.000755$ $0.000755$ $0.003079$ $(13,4)$ $0.001823$ $0.000521$ $0.000694$ $0.002282$ $(13,5)$ $0.001589$ $0.000388$ $0.000644$ $0.001804$ $(13,6)$ $0.001378$ $0.000303$ $0.000602$ $0.001485$ $(13,7)$ $0.001181$ $0.000245$ $0.000566$ $0.001256$ $(13,8)$ $0.000993$ $0.000204$ $0.000533$ $0.001085$ $(13,9)$ $0.000810$ $0.000173$ $0.000505$ $0.000950$ $(13,10)$ $0.000628$ $0.000148$ $0.000478$ $0.000843$ $(13,11)$ $0.000440$ $0.000129$ $0.000454$ $0.000755$ $(13,12)$ $0.000237$ $0.000114$ $0.000432$ $0.000681$ $(14,1)$ $0.002720$ $0.002720$ $0.000907$ $0.009486$ $(14,2)$ $0.002324$ $0.001199$ $0.000799$ $0.004705$ $(14,3)$ $0.002026$ $0.000725$ $0.000725$ $0.003102$ $(14,4)$ $0.001780$ $0.000501$ $0.000667$ $0.002307$ $(14,5)$ $0.001567$ $0.000373$ $0.000621$ $0.001827$ $(14,6)$ $0.001375$ $0.000292$ $0.000581$ $0.001511$ $(14,7)$ $0.001198$ $0.000237$ $0.000547$ $0.001252$ $(14,8)$ $0.001031$ $0.000197$ $0.000517$ $0.001106$ $(14,9)$ $0.000870$ $0.000167$ $0.000490$ $0.000974$ $(14,10)$ $0.000712$ $0.000144$ $0.000466$ $0.000865$ $(14,11)$ $0.000552$ $0.000125$ $0.000443$ $0.000768$ $(14,12)$ $0.000387$ $0.000111$ $0.000422$ $0.000699$ $(14,13)$ $0.000209$ $0.000099$ $0.000403$ $0.000635$
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