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Table 3 Frequency distribution for the number of sign changes between equivalent Hadamard matrices

From: Extended families of 2D arrays with near optimal auto and low cross-correlation

p 0 1 2 3 4 5 6 7 8
19 0.367167 0.54284 0.067909 0.017541 0.003751 0.000664 0.00011 0.000017 0.000001
23 0.36764376 0.54392703 0.06669008 0.01737125 0.00362844 0.00063027 0.00009564 0.00001197 0.00000136
31 0.368055 0.545614 0.065292 0.016755 0.003587 0.000598 0.000084 0.000014 0.000001
71 0.367799 0.548824 0.063183 0.0162 0.003383 0.000518 0.000077 0.000015 0.000001
127 0.367731 0.550441 0.062346 0.015706 0.003159 0.000544 0.000061 0.000011 0.000001
199 0.36784095 0.55094903 0.06192199 0.01555615 0.00312504 0.0005226 0.00007395 0.00000912 0.00000103
307 0.367281 0.551637 0.061785 0.015528 0.003128 0.000551 0.000082 0.000007 0.000001
419 0.368032 0.551334 0.0616 0.015369 0.00306 0.000521 0.000072 0.000011 0.000001
499 0.367942 0.551026 0.061757 0.015476 0.003181 0.000536 0.000075 0.000007 0
  1. The sum of the entries on each row is normalized to 1. For p = 23, at level 2 there are 3 × 24 = 72 sign changes between equivalent Hadamard matrices and this happens for 6.67% of shuffles. Note that the most frequent result, for all p, is when the number of toggles is the same as for a parent to child shuffle (level 1, which happens in about 55% of all shuffles)