On nonsingularity of circulant matrices

In Communication theory and Coding, it is expected that certain circulant matrices having k ones and k+1 zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when 2k+1 is either a power of a prime, or a product of two distinct primes. For any other integer 2k+1 we construct circulant matrices having determinant 0. The smallest singular matrix appears when 2k+1=45. The possibility for such matrices to be singular is rather low, smaller than 10⁻⁴ in this case.