Economic system based on the theories of Karl Marx and Vladimir Lenin in which the means of production are owned by the government?

The answer to this specific question is Socialism, not Communism. According to Marx, after the revolution by the proletariat working class, the means of production would be seized by the workers and a government set up by the proletariat. This government would own all property. Eventually, Communism would evolve and there would be no government. Thus any situation where there even IS a government would not be Marx's vision of communism.