Residue AG code 

Description

Also called a differential code. 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 secret-sharing schemes [4].

Parents

Children

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.
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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/tree/main/codes/classical/q-ary_digits/ag/residueAG/residue.yml.