We prove that for all positive integer k and for all sufficiently small /spl epsiv/<0 if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2/sup em/ which for all inputs X/spl sube/{0, 1, ..., n-1} computes in time kn the parity of the number of elements of the set of all pairs (x,y) with the property x/spl isin/X, y/spl isin/X, x>y, x+y/spl isin/X. For the proof of this fact we show that if A=(/spl alpha//sub i,j/)/sub i=0, j=0//sup n/ is a random n by n matrix over the field with 2 elements with the condition that "/spl forall/, j, k, l/spl isin/{0, 1, ..., n-1}, i+j=k+l implies /spl alpha//sub i,j/=/spl alpha//sub k,l/" then with a high probability the rank of each /spl delta by /spl delta submatrix of A is at least c/spl delta/|log /spl delta/|/sup -2, where c<0 is an absolute constant and n is sufficiently large with respect to /spl delta/.
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