Multi-point codes from the GGS curves (Q2176293)

Algebraic curves over finite fields can be used to obtain error correcting codes since the seminal work of Goppa in the early 1980s. Algebraic geometric (AG) codes have ``good'' parameters when the underlying curve has many rational points with respect to its genus. For this reason, maximal curves (i.e. curves attaining the upper bound in the Hasse-Weil bound) have been widely investigated. Recently, AG codes from Hermitian, Suzuki, Klein quartic, GK, and GGS curves and their quotients attracted a lot of attention. Most of the constructions of AG codes are one-point. In the case of multi-point AG codes, the main problem is a suitable description of Riemann-Roch spaces associated with divisors having a large support. This paper deals with the construction of AG codes defined from GGS curves, a generalization of the GK curve. In particular, the authors describe bases for the Riemann-Roch spaces associated with some rational places, and characterize explicitly the Weierstrass semigroups and pure gaps (a generalization of gaps) by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. As a byproduct, multi-point codes with parameters achieving new records are obtained.