https://hal.inria.fr/hal-01275753Geil, OlavOlavGeilDepartment of Mathematical Sciences [Aalborg] - AAU - Aalborg University [Denmark]Martin, StefanoStefanoMartinDepartment of Mathematical Sciences [Aalborg] - AAU - Aalborg University [Denmark]Martínez-Peñas, UmbertoUmbertoMartínez-PeñasDepartment of Mathematical Sciences [Aalborg] - AAU - Aalborg University [Denmark]Matsumoto, RyutarohRyutarohMatsumotoTITECH - Tokyo Institute of Technology [Tokyo]Ruano, DiegoDiegoRuanoDepartment of Mathematical Sciences [Aalborg] - AAU - Aalborg University [Denmark]On asymptotically good ramp secret sharing schemesHAL CCSD2016[INFO.INFO-CR] Computer Science [cs]/Cryptography and Security [cs.CR]Tillich, Jean-PierrePascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich2016-02-18 10:03:472018-04-24 16:16:022016-02-19 10:43:20enConference papersapplication/pdf1Asymptotically good sequences of ramp secret sharing schemes have been intensively studied by Cramer et al. in [1,2,3,4,5,6,7,8]. In those works the focus is on full privacy and full reconstruction. We propose an alternative definition of asymptotically good sequences of ramp secret sharing schemes where a small amount of information leakage is allowed (and possibly also non-full recovery). By a non-constructive proof we demonstrate the existence of sequences that – following our definition of goodness – have parameters arbitrary close to the optimal ones. Moreover – still using our definition – we demonstrate how to concretely construct asymptotically good sequences of schemes from sequences of algebraic geometric codes related to a tower of function fields. Our study involves a detailed treatment of the relative generalized Hamming weights of the involved codes.