Running time worsening when matrices of size 2^k are involved.

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I have experienced a worsening in the performances of a software when data of size 2^k for some k is involved. The problem is very similar to the one exhibited in the following link:
the software I have implemented is more complex than just the svd command, yet the effect is very similar (even though it is more clear on matrices of size 1023,1024,1025 and 2047,2048,2049). I have run the code also on different computers with different OSs and I have tried to change the number of threads by using the command maxnumcompthreads, but none of these seemed to work. I am confident that the problem is related to the "cache resonance effect" that is cited in the answer in the link. I would like to know more about this in order to justify the results that I am getting.

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