Tensor Rank and Border Rank of Band Toeplitz Matrices

Let Bn,h,k be the class of n × n band Toeplitz matrices A = (ai,j ) such that ai,j = 0 if i - j k or j - i h, k h and Sn,k the subclass of Bn,k,k made up by symmetric matrices. We show that rkF (Bn,h,k ) = n + h - 1 if the field F contains a primitive $(n - 1 + h)$th root of unity, ${brk}F (Bn,h,k ) = n + k - 1,{rk}R (Sn,k ) = {brk}R (Sn,k ) = n + 2 {{(k - 1)} / 2} $ if ${ R}$ is the real field and $k {n / 2}$.