The local dynamics of 10 substituted polyacetylenes in the glassy state have been investigated using a quasielastic neutron scattering technique in a time range of picoseconds to several tens of picoseconds to see a relationship between the local mobility and the gas permeability of these polymers. Even in the glassy state, these polymers show quasielastic scattering components, suggesting that certain stochastic motions occur in the glassy state. The dynamic scattering laws S(Q,omega) of the quasielastic components were well fitted to the sum of two Lorentzians, i.e., the narrow (slow) and broad (fast) components. It was found that both the relaxation rate Gamma(n) and the fraction A of the narrow (slow) component show positive correlations with oxygen permeability coefficient (Po-2), suggesting that the local mobility of the matrix polymers plays an important role in gas permeability. We then defined local flux F, which is the product of Gamma(n) and A(n) as a measure of the local mobility to find that F is proportional to the diffusion coefficient of O-2 gas (Do(2)), i.e., F proportional to Do(2). To explain and discuss this relation, we have proposed a random gate model, where mobile side groups in the matrix polymer act as a gate for gas diffusion.