"If I understand what communism is, the businesses are actually run and controlled by the government."
No, that's socialism. Communism is where people are all equal in the fact that they all use their specialties and innate talents to contribute to the greater good. In a true communist society, there is no government. There may be people who speak for the group, but it is supposed to be speaking the general consensus.
Of course, as a realist you can see the issue with that, can't you?
What we currently call communist regimes are actually socialist, bordering on fascist. I don't think that the world has seen anything resembling the ideal of communism outside of the hippie communes that dotted the landscape in the 70's, or an abby or something of the sort.