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
- Evaluation AG code — Any residue AG code of differential forms can be equivalently stated as an evaluation AG code of functions ([1], Lemma 15.3.10; [5], Thm. 2.72). In addition, evaluation and residue AG codes are dual to each other ([1], pg. 313; [5]).
- Algebraic-geometry (AG) code
Children
References
- [1]
- W. C. Huffman, J.-L. Kim, and P. Solé, 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.
Page edit log
- Victor V. Albert (2022-08-11) — most recent
- Victor V. Albert (2022-03-21)
Cite as:
“Residue AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/residue