In this note the kernel \(K_ A\) of biinfinite banded block Toeplitz matrices \[ \begin{pmatrix} \ldots & 0 & A_0 & A_1 & \ldots & A_ p & \ldots \\ \ldots & 0 & A_0 & A_1 & \ldots & A_ p & \ldots \end{pmatrix} \] is described. To this end an isomorphism between \(K_ A\) and a \({\mathbb{C}}[z]\)-module \(K_ A\) is introduced. It is finitely generated. The theorem in this note states that the following are equivalent: (i) A has full column rank, (ii) dim \(K_ A<\infty\), (iii) \(\bar K_ A\) is a torsion module. Moreover dim \(K_ A\) equals the degree of a characteristic polynomial of A. The theorem generalizes a result of P. W. Smith and H. Wolkowicz [Linear Algebra Appl. 57, 115-130 (1984)].