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Constant dimension codes from Riemann-Roch spaces

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).
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  • 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.

    Mathematics Subject Classification: Primary: 51E21, 51E22; Secondary: 94B05.

    Citation:

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  • 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}}$
     | Show Table
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    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}$
     | Show Table
    DownLoad: CSV

    Table 3.  Rates of $\mathcal{H}_{k,s}$, $\mathcal{A}_{k,s}$, $\mathcal{B}_{k,s,w}$, $\mathcal{C}_{k,s,w}$ for $q=16$, $8\leq n\leq 14$, $1\leq s <n$, $w=3$, $k=5$

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