Let G be a 2k-edge-connected graph with K >= 0 and let L(upsilon) {k,...,d(G)(upsilon)} for every upsilon epsilon V (G). A spanning subgraph F of G is called an L-factor, if d(F) (upsilon) epsilon L(upsilon) for every upsilon epsilon V (G). In this article, we show that if vertical bar L(upsilon)vertical bar >= inverted left perpendicular d(G)(upsilon)/2 inverted right perpendicular + 1 for every upsilon epsilon V(G), then G has a k-edge-connected L-factor. We also show that if k >= 1 and L(upsilon) = {inverted left perpendicular d(G)(upsilon)/2 inverted right perpendicular,..., inverted left perpendicular d(G)(upsilon)/2 inverted right perpendicular + k} for every upsilon epsilon V(G), then G has a k-edge-connected L-factor.