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I imagine that exhaustive color look up tables need to be 256 x 256 x 256 = 4096 x 4096 pixels.

Is that correct?

If so, how are these color look up tables (Hald CLUT) only 1600 x 1600 pixels?

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The lookup tables does not necessarily need to cover the whole colour space. According to RawPedia values missing in the LUT will simply be interpolated.

From the linked page:

Even if your HaldCLUT image contains colors in only 8-bit precision, missing values will be interpolated so that posterization will not occur in your photo. As such, since there is no visual loss of quality, we recommend using the level 12 identity file, or even an 8-level one

When creating a 12 level LUT I get 1728x1728 pixels which is close to the 1600x1600 of the ones you linked to.

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