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

Constant dimension codes from Riemann-Roch spaces

Daniele Bartoli 1, , Matteo Bonini 2, and  Massimo Giulietti 1,

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.

Keywords: Constant dimension codes, Riemann-Roch spaces, Random Network Codes.
Mathematics Subject Classification: Primary: 51E21, 51E22; Secondary: 94B05.
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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd edition, 1994. Google Scholar

[8]

J. P. Hansen, Riemann-Roch theorem and linear network codes, Int. J. Math. Comput. Sci., 10 (2015), 1-11. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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, 2010,405-408Google Scholar

show all references

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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd edition, 1994. Google Scholar

[8]

J. P. Hansen, Riemann-Roch theorem and linear network codes, Int. J. Math. Comput. Sci., 10 (2015), 1-11. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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, 2010,405-408Google Scholar

Table 1.  Normalized weight, rate, and normalized minimal distance, $s>1$
Normalized weightRateNormalized 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}}$
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Normalized weightRateNormalized 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}}$
Table 2.  Normalized weight, rate, and normalized minimal distance for $g=1$, $s>1$
Normalized weightRateNormalized 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}$
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Normalized weightRateNormalized 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}$
Table 3.  Rates of $\mathcal{H}_{k,s}$, $\mathcal{A}_{k,s}$, $\mathcal{B}_{k,s,w}$, $\mathcal{C}_{k,s,w}$ for $q=16$, $8\leq n\leq 14$, $1\leq s <n$, $w=3$, $k=5$
$(n,s)$ $\mathcal{H}_{k,s}$ $\mathcal{A}_{k,s}$ $\mathcal{B}_{k,s,w}$ $\mathcal{C}_{k,s,w}$
$(8,1)$ $0.003750$ $0.003750$ $0.001250$ $0.008732$
$(8,2)$ $0.003005$ $0.001616$ $0.001077$ $0.004286$
$(8,3)$ $0.002420$ $0.000959$ $0.000959$ $0.002802$
$(8,4)$ $0.001915$ $0.000654$ $0.000868$ $0.002058$
$(8,5)$ $0.001452$ $0.000481$ $0.000792$ $0.001611$
$(8,6)$ $0.001002$ $0.000373$ $0.000728$ $0.001311$
$(8,7)$ $0.000536$ $0.000300$ $0.000671$ $0.001095$
$(9,1)$ $0.003522$ $0.003522$ $0.001174$ $0.008931$
$(9,2)$ $0.002872$ $0.001526$ $0.001017$ $0.004394$
$(9,3)$ $0.002368$ $0.000909$ $0.000909$ $0.002880$
$(9,4)$ $0.001938$ $0.000622$ $0.000826$ $0.002121$
$(9,5)$ $0.001551$ $0.000459$ $0.000758$ $0.001665$
$(9,6)$ $0.001184$ $0.000357$ $0.000700$ $0.001360$
$(9,7)$ $0.000821$ $0.000287$ $0.000649$ $0.001141$
$(9,8)$ $0.000440$ $0.000237$ $0.000604$ $0.000976$
$(10,1)$ $0.003322$ $0.003322$ $0.001107$ $0.009093$
$(10,2)$ $0.002746$ $0.001445$ $0.000964$ $0.004482$
$(10,3)$ $0.002302$ $0.000865$ $0.000865$ $0.002943$
$(10,4)$ $0.001929$ $0.000593$ $0.000788$ $0.002173$
$(10,5)$ $0.001595$ $0.000439$ $0.000726$ $0.001710$
$(10,6)$ $0.001286$ $0.000341$ $0.000673$ $0.001400$
$(10,7)$ $0.000987$ $0.000275$ $0.000627$ $0.001178$
$(10,8)$ $0.000686$ $0.000228$ $0.000586$ $0.001011$
$(10,9)$ $0.000369$ $0.000192$ $0.000549$ $0.000880$
$(11,1)$ $0.003145$ $0.003145$ $0.001048$ $0.009229$
$(11,2)$ $0.002628$ $0.001374$ $0.000916$ $0.004555$
$(11,3)$ $0.002232$ $0.000824$ $0.000824$ $0.002996$
$(11,4)$ $0.001901$ $0.000566$ $0.000754$ $0.002215$
$(11,5)$ $0.001609$ $0.000420$ $0.000697$ $0.001746$
$(11,6)$ $0.001341$ $0.000327$ $0.000648$ $0.001433$
$(11,7)$ $0.001087$ $0.000264$ $0.000606$ $0.001209$
$(11,8)$ $0.000837$ $0.000219$ $0.000568$ $0.001040$
$(11,9)$ $0.000584$ $0.000185$ $0.000534$ $0.000908$
$(11,10)$ $0.000314$ $0.000159$ $0.000503$ $0.000802$
$(12,1)$ $0.002987$ $0.002987$ $0.000996$ $0.009343$
$(12,2)$ $0.002518$ $0.001309$ $0.000873$ $0.004617$
$(12,3)$ $0.002161$ $0.000788$ $0.000788$ $0.003040$
$(12,4)$ $0.001865$ $0.000542$ $0.000722$ $0.002251$
$(12,5)$ $0.001605$ $0.000403$ $0.000669$ $0.001778$
$(12,6)$ $0.001368$ $0.000315$ $0.000624$ $0.001461$
$(12,7)$ $0.001146$ $0.000254$ $0.000585$ $0.001234$
$(12,8)$ $0.000932$ $0.000211$ $0.000551$ $0.001064$
$(12,9)$ $0.000720$ $0.000179$ $0.000519$ $0.000931$
$(12,10)$ $0.000504$ $0.000154$ $0.000491$ $0.000824$
$(12,11)$ $0.000272$ $0.000134$ $0.000465$ $0.000736$
$(13,1)$ $0.002846$ $0.002846$ $0.000949$ $0.009440$
$(13,2)$ $0.002417$ $0.001251$ $0.000834$ $0.004670$
$(13,3)$ $0.002092$ $0.000755$ $0.000755$ $0.003079$
$(13,4)$ $0.001823$ $0.000521$ $0.000694$ $0.002282$
$(13,5)$ $0.001589$ $0.000388$ $0.000644$ $0.001804$
$(13,6)$ $0.001378$ $0.000303$ $0.000602$ $0.001485$
$(13,7)$ $0.001181$ $0.000245$ $0.000566$ $0.001256$
$(13,8)$ $0.000993$ $0.000204$ $0.000533$ $0.001085$
$(13,9)$ $0.000810$ $0.000173$ $0.000505$ $0.000950$
$(13,10)$ $0.000628$ $0.000148$ $0.000478$ $0.000843$
$(13,11)$ $0.000440$ $0.000129$ $0.000454$ $0.000755$
$(13,12)$ $0.000237$ $0.000114$ $0.000432$ $0.000681$
$(14,1)$ $0.002720$ $0.002720$ $0.000907$ $0.009486$
$(14,2)$ $0.002324$ $0.001199$ $0.000799$ $0.004705$
$(14,3)$ $0.002026$ $0.000725$ $0.000725$ $0.003102$
$(14,4)$ $0.001780$ $0.000501$ $0.000667$ $0.002307$
$(14,5)$ $0.001567$ $0.000373$ $0.000621$ $0.001827$
$(14,6)$ $0.001375$ $0.000292$ $0.000581$ $0.001511$
$(14,7)$ $0.001198$ $0.000237$ $0.000547$ $0.001252$
$(14,8)$ $0.001031$ $0.000197$ $0.000517$ $0.001106$
$(14,9)$ $0.000870$ $0.000167$ $0.000490$ $0.000974$
$(14,10)$ $0.000712$ $0.000144$ $0.000466$ $0.000865$
$(14,11)$ $0.000552$ $0.000125$ $0.000443$ $0.000768$
$(14,12)$ $0.000387$ $0.000111$ $0.000422$ $0.000699$
$(14,13)$ $0.000209$ $0.000099$ $0.000403$ $0.000635$
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$(n,s)$ $\mathcal{H}_{k,s}$ $\mathcal{A}_{k,s}$ $\mathcal{B}_{k,s,w}$ $\mathcal{C}_{k,s,w}$
$(8,1)$ $0.003750$ $0.003750$ $0.001250$ $0.008732$
$(8,2)$ $0.003005$ $0.001616$ $0.001077$ $0.004286$
$(8,3)$ $0.002420$ $0.000959$ $0.000959$ $0.002802$
$(8,4)$ $0.001915$ $0.000654$ $0.000868$ $0.002058$
$(8,5)$ $0.001452$ $0.000481$ $0.000792$ $0.001611$
$(8,6)$ $0.001002$ $0.000373$ $0.000728$ $0.001311$
$(8,7)$ $0.000536$ $0.000300$ $0.000671$ $0.001095$
$(9,1)$ $0.003522$ $0.003522$ $0.001174$ $0.008931$
$(9,2)$ $0.002872$ $0.001526$ $0.001017$ $0.004394$
$(9,3)$ $0.002368$ $0.000909$ $0.000909$ $0.002880$
$(9,4)$ $0.001938$ $0.000622$ $0.000826$ $0.002121$
$(9,5)$ $0.001551$ $0.000459$ $0.000758$ $0.001665$
$(9,6)$ $0.001184$ $0.000357$ $0.000700$ $0.001360$
$(9,7)$ $0.000821$ $0.000287$ $0.000649$ $0.001141$
$(9,8)$ $0.000440$ $0.000237$ $0.000604$ $0.000976$
$(10,1)$ $0.003322$ $0.003322$ $0.001107$ $0.009093$
$(10,2)$ $0.002746$ $0.001445$ $0.000964$ $0.004482$
$(10,3)$ $0.002302$ $0.000865$ $0.000865$ $0.002943$
$(10,4)$ $0.001929$ $0.000593$ $0.000788$ $0.002173$
$(10,5)$ $0.001595$ $0.000439$ $0.000726$ $0.001710$
$(10,6)$ $0.001286$ $0.000341$ $0.000673$ $0.001400$
$(10,7)$ $0.000987$ $0.000275$ $0.000627$ $0.001178$
$(10,8)$ $0.000686$ $0.000228$ $0.000586$ $0.001011$
$(10,9)$ $0.000369$ $0.000192$ $0.000549$ $0.000880$
$(11,1)$ $0.003145$ $0.003145$ $0.001048$ $0.009229$
$(11,2)$ $0.002628$ $0.001374$ $0.000916$ $0.004555$
$(11,3)$ $0.002232$ $0.000824$ $0.000824$ $0.002996$
$(11,4)$ $0.001901$ $0.000566$ $0.000754$ $0.002215$
$(11,5)$ $0.001609$ $0.000420$ $0.000697$ $0.001746$
$(11,6)$ $0.001341$ $0.000327$ $0.000648$ $0.001433$
$(11,7)$ $0.001087$ $0.000264$ $0.000606$ $0.001209$
$(11,8)$ $0.000837$ $0.000219$ $0.000568$ $0.001040$
$(11,9)$ $0.000584$ $0.000185$ $0.000534$ $0.000908$
$(11,10)$ $0.000314$ $0.000159$ $0.000503$ $0.000802$
$(12,1)$ $0.002987$ $0.002987$ $0.000996$ $0.009343$
$(12,2)$ $0.002518$ $0.001309$ $0.000873$ $0.004617$
$(12,3)$ $0.002161$ $0.000788$ $0.000788$ $0.003040$
$(12,4)$ $0.001865$ $0.000542$ $0.000722$ $0.002251$
$(12,5)$ $0.001605$ $0.000403$ $0.000669$ $0.001778$
$(12,6)$ $0.001368$ $0.000315$ $0.000624$ $0.001461$
$(12,7)$ $0.001146$ $0.000254$ $0.000585$ $0.001234$
$(12,8)$ $0.000932$ $0.000211$ $0.000551$ $0.001064$
$(12,9)$ $0.000720$ $0.000179$ $0.000519$ $0.000931$
$(12,10)$ $0.000504$ $0.000154$ $0.000491$ $0.000824$
$(12,11)$ $0.000272$ $0.000134$ $0.000465$ $0.000736$
$(13,1)$ $0.002846$ $0.002846$ $0.000949$ $0.009440$
$(13,2)$ $0.002417$ $0.001251$ $0.000834$ $0.004670$
$(13,3)$ $0.002092$ $0.000755$ $0.000755$ $0.003079$
$(13,4)$ $0.001823$ $0.000521$ $0.000694$ $0.002282$
$(13,5)$ $0.001589$ $0.000388$ $0.000644$ $0.001804$
$(13,6)$ $0.001378$ $0.000303$ $0.000602$ $0.001485$
$(13,7)$ $0.001181$ $0.000245$ $0.000566$ $0.001256$
$(13,8)$ $0.000993$ $0.000204$ $0.000533$ $0.001085$
$(13,9)$ $0.000810$ $0.000173$ $0.000505$ $0.000950$
$(13,10)$ $0.000628$ $0.000148$ $0.000478$ $0.000843$
$(13,11)$ $0.000440$ $0.000129$ $0.000454$ $0.000755$
$(13,12)$ $0.000237$ $0.000114$ $0.000432$ $0.000681$
$(14,1)$ $0.002720$ $0.002720$ $0.000907$ $0.009486$
$(14,2)$ $0.002324$ $0.001199$ $0.000799$ $0.004705$
$(14,3)$ $0.002026$ $0.000725$ $0.000725$ $0.003102$
$(14,4)$ $0.001780$ $0.000501$ $0.000667$ $0.002307$
$(14,5)$ $0.001567$ $0.000373$ $0.000621$ $0.001827$
$(14,6)$ $0.001375$ $0.000292$ $0.000581$ $0.001511$
$(14,7)$ $0.001198$ $0.000237$ $0.000547$ $0.001252$
$(14,8)$ $0.001031$ $0.000197$ $0.000517$ $0.001106$
$(14,9)$ $0.000870$ $0.000167$ $0.000490$ $0.000974$
$(14,10)$ $0.000712$ $0.000144$ $0.000466$ $0.000865$
$(14,11)$ $0.000552$ $0.000125$ $0.000443$ $0.000768$
$(14,12)$ $0.000387$ $0.000111$ $0.000422$ $0.000699$
$(14,13)$ $0.000209$ $0.000099$ $0.000403$ $0.000635$
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