Hermitian code[1][2]

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 \( \mathbb{F}_q[x]\).