Residue AG code 

Also known as 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].

Parents

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.

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.

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.