Hermitian code[1,2][3; Sec. 4.4.3]

Description

Evaluation AG code of rational functions evaluated on points lying on a Hermitian curve in either affine or projective space. Hermitian codes improve over RS codes in length: that RS codes have length at most \(q+1\) 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 \(q^3\), epicyclic codes have length \((q-2)(q+1)^2\), and projective coeds have length \(q^3+1\).

The affine-space equation defining a Hermitian curve over \(\mathbb{F}_q = GF(q)\) is \(H(x,y) = x^{q+1} + y^{q+1} - 1\), while the projective-space equation, or Fermat version, is \(H(x:y:z) = x^{q+1} + y^{q+1} - z^{q+1}\). 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.

To define the codes, 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} The Hermitian code \( C \) over is \begin{align} C = \{(f(\alpha_i))_{\alpha_i \in S}, \: f \in M_D \}~. \tag*{(3)}\end{align}

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 field trace in \(\mathbb{F}_{q^n}\), 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 \( \mathbb{F}_q[x]\).

There exists a family of Hermitian codes invariant under \(PGL(2,GF(q))\) [4].