On the Berlekamp/Massey algorithm and counting singular Hankel matrices over a finite field

We derive an explicit count for the number of singular n × n Hankel (Toeplitz) matrices whose entries range over a finite field with q elements by observing the execution of the Berlekamp/ Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n × n Toeplitz matrices with 0’s on the diagonal is q + q − q. We also derive the count for all n×n Hankel matrices of rank r with generic rank profile, i.e., whose first r leading principal submatrices are non-singular and the rest are singular, namely q r(q − 1)r in the case r < n and q(q − 1)r in the case r = n. This result generalizes to block-Hankel matrices as well.