Generate random binaries with avg hamming distance of 50%?

I want to generate binaries, where the average hamming distance between the items is 50%. The second condition is that the distance should not fall below 40% or go above 60%. Another complication is that the items are not generated in sequence, but once in a while and I don't want to loop over all of the items to check and regenerate, because it will become a slow process after a while.

Is there a mechanism or algorithm to achieve this ?

Currently I use the following code :

def rand(size):
    op = np.random.uniform()
    return np.random.choice([0,1], size=size, p=[op, 1-op] )

but it breaks even when I generate 10 items f.e. (hamming dist) :

[ [   0 2510 8209 4305 3896 1619 7231 6356 8103 3265]
  [2510    0 8131 4347 3940 1697 7219 6334 8037 3305]
  [8209 8131    0 5858 6449 9312 2100 3317 1030 7196]
  [4305 4347 5858    0 4661 4088 5598 5311 5764 4590]
  [3896 3940 6449 4661    0 3485 6093 5650 6385 4251]
  [1619 1697 9312 4088 3485    0 8034 6739 9152 2716]
  [7231 7219 2100 5598 6093 8034    0 3831 2238 6510]
  [6356 6334 3317 5311 5650 6739 3831    0 3405 5933]
  [8103 8037 1030 5764 6385 9152 2238 3405    0 7112]
  [3265 3305 7196 4590 4251 2716 6510 5933 7112    0]]

min: 1030 Avg distance : 0.470624%

btw. binaries are 10 000 bits.