These problems have many practical applications in communications and
electrical engineering. The objective is to construct a binary sequence S_i
of length n that minimizes the autocorrelations between bits. Each bit in the
sequence takes the value +1 or -1.
With non-periodic (or open) boundary
conditions,
the k-th autocorrelation, C_k
is defined to be sum(i,0,n-k-1, S_i * S_i+k).
With periodic (or cyclic) boundary
conditions, the k-th autocorrelation, C_k
is defined to be sum(i,0,n-1, S_i * S_((i+k)mod n)).
The aim is to minimize the sum of the
squares of these autocorrelations. That is,
to minimize E=sum(k,1,n-1,C_k^2).