## Description

Evaluation AG code of rational functions evaluated on points lying on a Hermitian curve \(H(x,y) = x^{q+1} + y^{q+1} - 1\) over \(\mathbb{F}_q = GF(q)\) in either affine or projective space. Hermitian codes directly improve over RS codes in the sense that RS codes have length at most \(q\) while Hermitian codes have length \(q^3 + 1\).

Hermitian codes of polynomials of total degree at most \(D\) can come in affine and epicyclic flavours, depending on whether the evaluations are over the affine plane or the bicyclic plane. The affine codes have length \(q^3 - q\) while epicyclic codes have length \((q-2)(q+1)^2\). More precisely, fix \(r, D\) and let \begin{align} M_D = \left\{f(x,y,z) = \sum_{i+j \leq D = D}a_{i,j}x^{i}y^{j}z^{D - (i+j)}\right\} \tag*{(1)}\end{align} be the message space of degree-\(D\) polynomials and \begin{align} S = \{(x:y:z) \in PG(2,q) \mid H(x:y:z) = 0 \}~, \tag*{(2)}\end{align} where \(H(x:y:z) = x^{q+1} + y^{q+1} - z^{q+1}\) is the homogenized Hermitian curve over the projective plane. The Hermitian code \( C \) over is \begin{align} C = \{(f(\alpha_i))_{\alpha_i \in S}, \: f \in M_D \}~. \tag*{(3)}\end{align}

The form \(H(u,v,w) = u^{q+1} + v^{q+1} - w^{q+1}\) is the Fermat version of the Hermitian curve. Substituting \(u = x+z\), \(v = x+y\), and \(w = x+y+z \) yields \(H(x,y,z) = x^{q+1} - y^{q}z - yz^{q} \), the Stichtenoth version of the curve. In affine coordinates, the Stichtenoth form of the curve is \begin{align} f(x,y) = x^{q+1} - y^{q} - y = N(x) - \text{tr}(y)~, \tag*{(4)}\end{align} where \(N(x) := x^{(q^{n}-1)/(q-1)}\) and \(\text{tr} := 1 + x^{q} + \ldots + x^{q^n}\) are the field norm and trace of \(GF(F_{q^n}\), respectively. The Fermat version can be written as \(H(u,v,w) = u\overline{u} + v\overline{v} - w\overline{w}\), where the conjugation map \(\overline{u} = u^{q}\) is an isomorphism of \(\mathbb{F}_q \). In fact, when the field of evaluations \(\mathbb{F}_{q^2}\) is viewed as a quadratic extension of \(\mathbb{F}_q\) then the conjugation map is an \(\mathbb{F}_q\)-isomorphism that permutes the roots of the quadratic irreducible polynomial used to generate \(\mathbb{F}_{q^2}\) from \( \mathbb{F}_q[x]\).

## Protection

## Rate

## Decoding

## Parents

- Generalized RS (GRS) code — Hermitian codes are concatenated GRS codes [6].
- Evaluation AG code — Hermitian codes are evaluation AG codes with \(\cal X\) being a Hermitian curve ([4], Ex. 2.74). This curve is maximal, meaning that Hermitian codes are evaluation AG codes with maximum possible length given a fixed genus.

## Cousins

- Group-algebra code — Some Hermitian codes are group-algebra codes [7][8; Remark 16.4.14].
- Hermitian-hypersurface code — Hermitian-hypersurface codes reduce to Hermitian codes of polynomials when the hypersurface is a curve.

## References

- [1]
- H. Stichtenoth, “�ber die Automorphismengruppe eines algebraischen Funktionenk�rpers von Primzahlcharakteristik”, Archiv der Mathematik 24, 527 (1973) DOI
- [2]
- H. Tiersma, “Remarks on codes from Hermitian curves (Corresp.)”, IEEE Transactions on Information Theory 33, 605 (1987) DOI
- [3]
- K. Yang and P. V. Kumar, “On the true minimum distance of Hermitian codes”, Lecture Notes in Mathematics 99 (1992) DOI
- [4]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
- [5]
- 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
- [6]
- T. Yaghoobian and I. F. Blake, “Hermitian codes as generalized Reed-Solomon codes”, Designs, Codes and Cryptography 2, 5 (1992) DOI
- [7]
- Hansen, Johan P. Group codes on algebraic curves. Universität zu Göttingen. SFB Geometrie und Analysis, 1987.
- [8]
- W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) 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