Parameters for Algebraic Geometric Codes
and Secret Sharing Schemes
Curve
Suzuki over \(\mathbb{F}_8\)
Suzuki over \(\mathbb{F}_{32}\)
Hermitian over \(\mathbb{F}_{16}\)
Hermitian over \(\mathbb{F}_{64}\)
Klein over \(\mathbb{F}_8\)
GK over \(\mathbb{F}_{64}\)
GK over \(\mathbb{F}_{729}\)
Bounds
minimum weight of coset
minimum distance of code
Cosets
\( \mathcal{C}_L(D,D-C) \backslash \mathcal{C}_L(D,D-C-P)\)
\(\mathcal{C}_\Omega(D,K+C) \backslash \mathcal{C}_\Omega(D,K+C+P) \)
Codes
\( \mathcal{C}_L(D,D-C)\text{ or } \mathcal{C}_\Omega(D,K+C) \)
Goppa Bound = \(\max\{0,\text{deg} C\} \)
\[D = \sum_{R \text{ rational point}} R - P - Q\]
Order Bound Methods
Duursma-Kirov
[ref]
Duursma-Park
[ref]
Beelen
[ref]
Floor Bound Methods
ABZ
[ref]
Güneri-Stichtenoth-Taşkın
[ref]
Lundell-McCullough
[ref]
Coset Point
\(P\)
\(Q\)
Degree and Q Value of C
??
≤
\(≤\text{deg C}≤\)
≤
??
??
≤
\(≤C_q≤\)
≤
??
Curve Info
Equation \( x^{5} = y^{4}+y \)
Genus \( g = 6 \)
Canonical Divisor \( K = 10P \sim 10Q \)
Rational Points \( = 65 \)
Curve Info
Equation \( x^{9}=y^8+y \)
Genus \( g = 28 \)
Canonical Divisor \( K = 54P \sim 54Q \)
Rational Points \( = 513 \)
Curve Info
Equation \( y^8 + y = x^2 (x^8+x) \)
Genus \( g = 14 \)
Canonical Divisor \( K = 26P \sim 26Q \)
Rational Points \( = 65 \)
Curve Info
Equation \( y^{32} + y = x^4 (x^{32}+x) \)
Genus \( g = 124 \)
Canonical Divisor \( K = 246P \sim 246Q \)
Rational Points \( = 1025 \)
Curve Info
Equation \( x^3y+y^3+x \)
Genus \( g = 3 \)
Canonical Divisor \( K = P + 3Q \)
Rational Points \( = 24 \)
Point \(P = \infty \)
Point \(Q = (0,0) \)
Curve Info
Equation \(x^{9}-y^8+y+(y^2+y)^3 \)
Genus \( g = 10 \)
Canonical Divisor \( K = 18P \)
Rational Points \( = 221 \)
Curve Info
Equation \(x^{28}-y^{27}+y+(y^3+y)^7 \)
Genus \( g = 99 \)
Canonical Divisor \( K = 196P \)
Rational Points \( = 6058 \)
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