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×nn×n Hankel (Toeplitz) matrices whose entries range over a finite field with qq 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×nn×n Toeplitz matrices with 0’s on the diagonal is q2n−3+qn−1−qn−2q2n−3+qn−1−qn−2.We also derive the count for all n×nn×n Hankel matrices of rank rr with generic rank profile, i.e., whose first rr leading principal submatrices are non-singular and the rest are singular, namely qr(q−1)rqr(q−1)r in the case r

► We count singular square Hankel matrices over a finite field with some entries fixed. Entries may be fixed above or on, or equivalently below or on, the anti-diagonal.
► We count by executing the Berlekamp/Massey algorithm on the matrix entries.
► We also count singular square block-Hankel matrices with generic rank profile.