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Residue AG code

Alternative names: Differential code.

Description

Linear \(q\)-ary code defined using a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) and a linear space \(\Omega\) of certain rational differential forms \(\omega\).

Codewords are evaluations of residues of the differential forms in the specified points, \begin{align} \left(\text{Res}_{P_{1}}(\omega),\text{Res}_{P_{2}}(\omega),\cdots,\text{Res}_{P_{n}}(\omega)\right)\quad\quad\forall\omega\in\Omega~. \tag*{(1)}\end{align} The code is denoted as \(C_{\Omega}({\cal X},{\cal P},D)\), where the divisor \(D\) determines which rational rational differential forms to use.

Protection

Riemann-Roch theorem yields code length \(n\), dimension \(k\), and a lower bound on distance in terms of features of \(L\) and genus of the curve \(\cal X\) [1; Corr. 15.3.13]. Distance bounds can also be derived from how an algebraic curve \(\cal X\) is embedded in the ambient projective space [2].

Realizations

Improvements over the McEliece public-key cryptosystem to linear AG codes on curves of arbitrary genus [3]. Only the subfield subcode proposal remains resilient to attacks [1; Sec. 15.7.5.3].Algebraic geometric secret-sharing schemes [4].

Cousin

  • Cartier code— Every Cartier code is contained in a subfield subcode of a residue AG code. Cartier codes share similar asymptotic properties as subfield subcodes of residue AG codes, with both families admitting sequences of codes that achieve the GV bound.

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation code
Evaluation AG codeAG ECC
Parents
Evaluation AG code
Any residue AG code of differential forms can be equivalently stated as an evaluation AG code of functions [6,7][1; Lemma 15.3.10][5; Thm. 2.72]. In addition, evaluation and residue AG codes are dual to each other [5][1; pg. 313].
Algebraic-geometry (AG) codeECC
Residue AG code
Children
Goppa code
Tsfasman-Vladut-Zink (TVZ) code
TVZ codes are evaluation AG codes where \(\cal X\) is either a Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve, but can also be formulated as residue AG codes.

References

[1]
A. Couvreur, H. Randriambololona, "Algebraic Geometry Codes and Some Applications." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[2]
A. Couvreur, “The dual minimum distance of arbitrary-dimensional algebraic–geometric codes”, Journal of Algebra 350, 84 (2012) arXiv:0905.2345 DOI
[3]
H. Janwa and O. Moreno, Designs, Codes and Cryptography 8, 293 (1996) DOI
[4]
H. Chen and R. Cramer, “Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields”, Lecture Notes in Computer Science 521 (2006) DOI
[5]
T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
[6]
M. Tsfasman, S. Vlǎduţ, and D. Nogin. Algebraic geometric codes: basic notions. Vol. 139. American Mathematical Society, 2022.
[7]
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
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Zoo Code ID: residue

Cite as:
“Residue AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/residue
BibTeX:
@incollection{eczoo_residue, title={Residue AG code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/residue} }
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Cite as:

“Residue AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/residue

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/ag/residueAG/residue.yml.