A counter-example to Singer's conjecture for the algebraic transfer

Write $P_k:= \mathbb F_2[x_1,x_2,\ldots ,x_k]$ for the polynomial algebra over the prime field $\mathbb F_2$ with two elements, in $k$ generators $x_1, x_2, \ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra, $\mathcal A$. Let $GL_