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

Abstract

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 q2n−3+qn−1−qn−2.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 qr(q−1)r in the case r<n and qr−1(q−1)r in the case r=n. This result generalizes to block-Hankel matrices as well.