American Institute of Mathematical Sciences

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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.

[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

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.

[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

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}}$
Table Options

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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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