Symmetric Pseudo-Random Matrices

We consider the problem of generating symmetric pseudo-random sign (+/- 1) matrices based on the similarity of their spectra to Wigner's semicircular law. Using binary m-sequences (Golomb sequences) of lengths n = 2(m) - 1, we give a simple explicit construction of circulant n x n sign matrices and show that their spectra converge to the semicircular law when n grows. The Kolmogorov complexity of the proposed matrices equals to that of Golomb sequences and is at most 2log(2)(n) bits.