Bases for Riemann-Roch spaces of linearized function fields with applications to generalized algebraic geometry codes (Q6618625)

In coding theory, the computation of the dimension of Riemann-Roch spaces associated to divisors of a function field plays a fundamental role in determining the values of certain parameters of algebraic geometry codes (or Goppa codes).\N\NIn this paper, the author explicitly determines a basis and the dimension of certain Riemann-Roch spaces associated to divisors of the function field \({\mathbb F}_{q^n}(x, y)\) defined by the equation\N\[\NL(y)=h(x),\N\]\Nwhere \(L(y)=\sum_{i=0}^{r}a_iy^{q^i}\in{\mathbb F}_{q^n}[y]\) is a separable linearized polynomial with \(q^r\) roots in \({\mathbb F}_{q^n}\) and \(h(x)\in {\mathbb F}_{q^n}(x)\). As a consequence, generalized algebraic geometry codes with good parameters are constructed over such function fields.