## Description

Let \( G(x) \) be a polynomial describing a projective-plane curve with coefficients from \( GF(q^m) \) for some fixed integer \(m\). Let \( L \) be a finite subset of the extension field \( GF(q^m) \) where \(q\) is prime, meaning \( L = \{\alpha_1, \cdots, \alpha_n\} \) is a subset of nonzero elements of \( GF(q^m) \). A Goppa code \( \Gamma(L,G) \) is an \([n,k,d]_q\) linear code consisting of all vectors \(a = a_1, \cdots, a_n\) such that \( R_a(x) =0 \) modulo \(G(x)\), where \( R_a(x) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i} \).

Goppa codes are residue AG codes ([4], Thm. 15.3.28). Their duals are evaluation codes that are sometimes called geometric Reed Solomon codes ([5], Thm. 2.71).

## Protection

## Decoding

## Realizations

## Notes

## Parents

- Generalized RS (GRS) code — Goppa codes are \(GF(q)\)-subfield subcode of the dual of the GRS code over \(GF(q^m)\) with evaluation points \(\alpha_i\) and factors \(v_i=G(\alpha_i)^{-1}\) ([15], pg. 523; [5]).
- Cartier code — Goppa codes are Cartier codes from a curve of genus zero [16].

## Cousins

- \(q\)-ary linear LTC — Goppa codes are locally testable [17].
- Bose–Chaudhuri–Hocquenghem (BCH) code — Narrow-sense BCH codes are Goppa codes with \(L=\{1,\alpha^{-1},\cdots,\alpha^{1-n}\}\) and \(G(x)=x^{\delta-1}\) ([15], pg. 522).
- Binary quantum Goppa code — Classical Goppa codes are used to construct their quantum versions.

## References

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- E. Berlekamp, “Goppa codes”, IEEE Transactions on Information Theory 19, 590 (1973) DOI
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- W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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- N. Patterson, “The algebraic decoding of Goppa codes”, IEEE Transactions on Information Theory 21, 203 (1975) DOI
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- Daniel J. Bernstein, "Understanding binary-Goppa decoding." Cryptology ePrint Archive (2022).
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- V. M. SIDELNIKOV and S. O. SHESTAKOV, “On insecurity of cryptosystems based on generalized Reed-Solomon codes”, Discrete Mathematics and Applications 2, (1992) DOI
- [15]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
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- A. Couvreur, “Codes and the Cartier Operator”, (2012) arXiv:1206.4728
- [17]
- T. Kaufman and S. Litsyn, “Almost Orthogonal Linear Codes are Locally Testable”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) DOI

## Page edit log

- Victor V. Albert (2022-08-21) — most recent
- Victor V. Albert (2021-12-15)
- Manasi Shingane (2021-12-14)

## Cite as:

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