Algebraic geometric secret sharing schemes over large fields are asymptotically threshold

In Chen-Cramer Crypto 2006 paper \cite{cc} algebraic geometric secret sharing schemes were proposed such that the Fundamental Theorem in Information-Theoretically Secure Multiparty Computation by Ben-Or, Goldwasser and Wigderson \cite{BGW88} Chaum, Crepeau Damgard \cite{CCD88} can be established over constant-size base finite fields. These defined a curve of genus $g$ constant size field ${\bf F}_q$ is quasi-threshold following sense, any subset $u \leq T-1$ players (non qualified) has no information \geq T+2g$ (qualified) reconstruct secret. It natural to ask how far from threshold these are? How many subsets \in [T, T+2g-1]$ recover or have secret? In this it proved almost all [T,T+g-1]$ [T+g,T+2g-1]$ when $q$ goes infinity satisfies $\lim \frac{g}{\sqrt{q}}=0$. Then large fields are asymptotically case. We also analyze case fixed infinity.