  Recently, enlarged Krylov subspace methods, that consists of enlarging the
Krylov subspace by a maximum of t vectors per iteration based on the domain
decomposition of the graph of A, were introduced in the aim of reducing
communication when solving systems of linear equations Ax=b. In this paper, the
s-step enlarged Krylov subspace Conjugate Gradient methods are introduced,
whereby s iterations of the enlarged Conjugate Gradient methods are merged in
one iteration. The numerical stability of these s-step methods is studied, and
several numerically stable versions are proposed. Similarly to the enlarged
Krylov subspace methods, the s-step enlarged Krylov subspace methods have a
faster convergence than Krylov methods, in terms of iterations. Moreover, by
computing st basis vectors of the enlarged Krylov subspace
$\mathscr{K}_{k,t}(A,r_0)$ at the beginning of each s-step iteration,
communication is further reduced. It is shown in this paper that the introduced
methods are parallelizable with less communication, with respect to their
corresponding enlarged versions and to Conjugate Gradient.
