Syndrome Generation and Error Location Search for the Decoding of Algebraic-Geometry Codes on Plane Garcia–Stichtenoth Curves

In this paper, the basic function field of an asymptotically optimal tower of function fields by Garcia and Stichtenoth is studied, where bases of one-point linear systems can be explicitly constructed. A representation of finite GF(q
²
)-rational points on the associated plane curve is derived in this paper. This representation is exploited to extend the use of Horner's loops and the mechanism of Chien search in the decoding of Reed-Solomon codes for syndrome generation and error-location search, respectively, to algebraic-geometry codes built on the plane Garcia-Stichtenoth curve.