This is actually an area of heated debate right now. Short answer: we're not sure. There's a few competing models(with varying degrees of support) for how exactly molecular recognition for smell works. Our understanding of how much the "smell space" can be reduced and quantized will be influenced heavily by which of these models most closely reflects reality.
A paper from 2014 claimed that we could distinguish upwards of a trillion smells. This was refuted last year, with a paper claiming the original one was flawed. It's hard to know at this point if we might reduce the "smell space" to a small subset of base smells or not, there just isn't clear enough data yet.
One important point is that our genome has about 3% of our genes devoted to smell receptors - about 1000 genes. At least 400 of these are functional. This means that there are at least 400 different permutations of responses that could happen, and different receptors could react to different molecules in different ways, so the complexity is already huge. Combine multiple receptors simultaneously, it gets even more complex.
At the same time, that could be a lot of overkill and we might only use a subset of those receptors well, and there might be a much greatly reduced pool of "dimensions of odour space".
In short, we're not sure, it's a bit of a tangled mess, but we'll know a lot more in a few years.