Improved evaluation codes defined by plane valuations (Q982483)

The authors study the improved evaluation codes, introduced by \textit{H. E. Andersen} [Finite Fields Appl. 14, No. 1, 92--123 (2008; Zbl 1136.94010)], for the case of finitely generated order structure given by plane valuations [see, \textit{O. Zariski} and \textit{P. Samuel}, Commutative algebra. Vol. II. (The University Series in Higher Mathematics.) Princeton, N.J.-Toronto- London-New York: D. Van Nostrand Company, Inc. (1960; Zbl 0121.27801); \textit{M. Spivakovsky}, Am. J. Math. 112, No. 1, 107--156 (1990; Zbl 0716.13003)]. In Section 2, some generalities about plane valuation are exposed. Section 3 is the main of the paper and is devoted to study the parameters (length, dimension, and minimum distance) of the improved evaluation codes. Section 4 provides with a comparing example of improved evaluation codes and Section 5 describes, in a detailed manner, the minimal generating sets of the value semigroups of those valuations that are used in the paper.