This is a really difficult question because we simply don't know what a 'Communist' economy is. Nobody has ever defined it beyond a few beyond the very superficial level, like 'no state, no money, no property' or 'From each according to his ability, to each according to his need.'. It is not clear what this means.

Now, another question is weather their economy was socialist, the pre-stage to a eventual transition to communism. On this question people are divided, socialist well tell you that it was not socialist but rather 'state capitalists', while most everybody else will tell you that it was 'socialist'. Now all of this definition arguing is a bit pointless if you want to actually learn what the Soviet economy was about, so lets talk about that.

Because Marx never clearly state how these things should work, people came up with lots of different ideas. One very popular interpretation was that the economy should be completely owned and controlled by the state, while the state represented worker interests. The state would rationally plan the economy for the benefit of all and because that would avoid profits, it would be more efficient then capitalism.

People like Lenin and Bukharin wanted this sort of state driven system and they did implemented soon after they came to power, they took control over all major industries to implement this vision. I don't want to go into much detail about the whole history of this but essentially after that the the Soviet state controlled the majority of all industry. Stalin 'finished the job' when he collectivized agriculture in the 30s.

However, they never managed to really end private ownership and market, specially in small shops and productions. This accounted a significant contribution to the Soviet economy.

There was of course still quasi-property rights and illegal market to solve the many issues that this centrally planed economy was causing but the Soviet Union remained a Socialist Command and Control economy until the end.