Codes from Riemann-Roch spaces for \(y^{2} = x^{p} - x\) over \(\text{GF}(p)\) (Q622791)

The paper computes explicit basis for the vector spaces \(\mathcal{L}(D)\), \(D\) a divisor on the hyperelliptic curve \(\mathcal{X}: y^2=x^p-x\), defined over a field of characteristic \(p\), \(D\) invariant by the automorphism group \(G\) of \(\mathcal{X}\) and applies the results to the construction of AG codes deduced from those spaces. Section 2 studies the curve \(\mathcal{X}\) over the prime field \(\mathbb{F}_p\) and over its quadratic extension \(\mathbb{F}_{p^2}\), as well as the group \(G\) and the \(G\)-invariant divisors: \(D_1 =\sum P;\,\, P\in \mathcal{X}(\mathbb{F}_p) \), \(D_2 =\sum P;\,\, P\in \mathcal{X}(\mathbb{F}_{p^2})-\mathcal{X}(\mathbb{F}_p)\) and their integer lineal combinations. Then the paper deduces the wanted basis. Section 3 discusses the AG codes \(\mathcal{C}=\mathcal{C}(rD_1, D_2\), where \(r\geq 1\) is an integer, Proposition 2 giving the parameters \([n,k,d]\) of these codes, while Section 4 shows other examples obtained with the aid of the computational packages GAP and SAGE. Finally Section 5 suggests to extend the study to the curves \(y^m=x^p-x\), where \(m\) is a proper divisor of \(p+1\).