Hermitian code[1,2][3; Sec. 4.4.3]
Alternative names: Hermitian AG code.
Description
Evaluation AG code of rational functions on a Hermitian curve over \(\mathbb{F}_{q^2}\).
A standard affine model of the curve is \begin{align} y^q+y=x^{q+1}~, \tag*{(1)}\end{align} and this curve has \(q^3+1\) rational points and genus \(g=q(q-1)/2\).
Hermitian codes are usually defined as one-point AG codes \(C_L(D,mP_\infty)\), where \(D\) is the sum of the \(q^3\) affine rational points and \(P_\infty\) is the unique point at infinity. They provide much longer code lengths than Reed-Solomon codes over the same alphabet, since the standard one-point construction has length \(q^3\) over \(\mathbb{F}_{q^2}\).
There exists a family of Hermitian codes with automorphism group \(PGL(2,\mathbb{F}_q)\) [4].
Protection
Distance is determined by the divisor and the geometry of the Hermitian curve [5]; see [6; Sec. 5.3] for the general result. For the standard one-point code \(C_L(D,mP_\infty)\) of length \(n=q^3\), the designed distance satisfies \(d \geq n-m\).Rate
For the standard one-point code \(C_L(D,mP_\infty)\) with genus \(g=q(q-1)/2\), Riemann-Roch gives \begin{align} k=m-g+1 \tag*{(2)}\end{align} whenever \(2g-2 < m < n=q^3\).Decoding
Unique decoding using syndromes and error locator ideals for polynomial evaluations. Note that Hermitian codes are linear codes so we can compute the syndrome of a received vector. Moreover, akin to the error-locator ideals found in decoding RS codes, for the multivariate case we must define an error locator ideal \(\Lambda \) such that the variety of this ideal over \(\mathbb{F}^{2}_q\) is exactly the set of errors. The Sakata algorithm uses these two ingredients to get a unique decoding procedure [7].Notes
Certain structured classical optimization problems can be mapped into decoding and list decoding Hermitian codes via the Decoded Quantum Interferomentry (DQI) algorithm [8].Cousins
- Generalized RS (GRS) code— Hermitian codes are concatenated GRS codes [9].
- Group-algebra code— Some Hermitian codes are group-algebra codes [10][11; Remark 16.4.14].
- Hermitian-hypersurface code— Hermitian-hypersurface codes reduce to Hermitian codes of polynomials when the hypersurface is a curve.
- Tamo-Barg-Vladut code— Tamo-Barg-Vladut codes can be defined on Hermitian curves.
- Quantum Hermitian AG code
Primary Hierarchy
Parents
Hermitian codes are evaluation AG codes with \(\cal X\) being a Hermitian curve [2][6; Exam. 2.74]. This curve is maximal, meaning that Hermitian codes are evaluation AG codes with maximum possible length given a fixed genus. They are a special case of norm-trace codes [12].
Hermitian code
References
- [1]
- H. Tiersma, “Remarks on codes from Hermitian curves (Corresp.)”, IEEE Transactions on Information Theory 33, 605 (1987) DOI
- [2]
- R. E. Blahut, Algebraic Codes on Lines, Planes, and Curves (Cambridge University Press, 2001) DOI
- [3]
- M. Tsfasman, S. Vlǎduţ, and D. Nogin. Algebraic geometric codes: basic notions. Vol. 139. American Mathematical Society, 2022.
- [4]
- G. Korchmáros and P. Speziali, “Hermitian codes with automorphism group isomorphic to PGL(2,q) with q odd”, Finite Fields and Their Applications 44, 1 (2017) DOI
- [5]
- K. Yang and P. V. Kumar, “On the true minimum distance of Hermitian codes”, Lecture Notes in Mathematics 99 (1992) DOI
- [6]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
- [7]
- S. Sakata, “Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array”, Journal of Symbolic Computation 5, 321 (1988) DOI
- [8]
- A. Gu and S. P. Jordan, “Algebraic Geometry Codes and Decoded Quantum Interferometry”, (2025) arXiv:2510.06603
- [9]
- T. Yaghoobian and I. F. Blake, “Hermitian codes as generalized Reed-Solomon codes”, Designs, Codes and Cryptography 2, 5 (1992) DOI
- [10]
- Hansen, Johan P. Group codes on algebraic curves. Universität zu Göttingen. SFB Geometrie und Analysis, 1987.
- [11]
- W. Willems, “Codes in Group Algebras.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [12]
- O. Geil, “On codes from norm–trace curves”, Finite Fields and Their Applications 9, 351 (2003) DOI
Page edit log
- Victor V. Albert (2022-08-09) — most recent
- Shashank Sule (2022-04-03)
Cite as:
“Hermitian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hermitian