Constant dimension codes from Riemann-Roch spaces

Daniele Bartoli; Matteo Bonini; Massimo Giulietti

doi:10.3934/amc.2017051

Article Contents

2017, Volume 11, Issue 4: 705-713. Doi: 10.3934/amc.2017051

This issue Previous Article On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$ Next Article An active attack on a distributed Group Key Exchange system

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: July 2018

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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Abstract

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.

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

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Table 1. 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}}$
$\mathcal{A}_{k,s}$	$\frac{ks+1-g}{nks+1-g}$	$\frac{\log_q \binom{n+s-1}{s}}{(nks+1-g)(ks+1-g)}$	$\frac{1}{s+\frac{1-g}{k}}$
$\mathcal{B}_{k,s,w}$	$\frac{ks+1-g}{nkw+1-g}$	$\frac{\log_q U_{n,s,0,w}}{(nkw+1-g)(ks+1-g)}$	$\frac{1}{s+\frac{1-g}{k}}$
$\mathcal{C}_{k,s,w}$	$\frac{ks+1-g}{nkw+1-g}$	$\frac{\log_q U^{\prime}_{n,s,s-w(n-1),w}}{(nkw+1-g)(ks+1-g)}$	$\frac{1}{s+\frac{1-g}{k}}$

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Table 2. 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}$
$\mathcal{A}_{k,s}$	$\frac{1}{s}$	$\frac{\log_q \binom{n+s-1}{s}}{nk^2s^2}$	$\frac{1}{s}$
$\mathcal{B}_{k,s,w}$	$\frac{s}{nw}$	$\frac{\log_q U_{n,s,0,w}}{nk^2ws}$	$\frac{1}{s}$
$\mathcal{C}_{k,s,w}$	$\frac{s}{nw}$	$\frac{\log_q U^{\prime}_{n,s,s-w(n-1),w}}{nk^2ws}$	$\frac{1}{s}$

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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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