Tire having at least one carcass reinforcement, a tread, a belt reinforcer having a first and a second end and: a reinforcing ply, having a transverse half-width L2, having a reinforcing polymer layer between two layers of coating rubber, at least one hooping ply, surmounting the reinforcing ply, having a transverse half-width L1, and filamentary reinforcing elements, parallel two by two, forming an angle at most equal to 7° with respect to the circumferential direction, the tire having at least one layer of edge uncoupling rubber, positioned at least radially between the hooping ply and the reinforcing ply and positioned at at least one transverse end of the belt reinforcer, the layer of edge uncoupling rubber having a total length between 10 and 40 mm and a thickness between 0.2 and 1.2 mm, having at least one reinforcing ply, and at at least one end of the belt reinforcer, the half-widths L1 and L2 are different, and manifest a difference (L1−L2) between 5 and 10 mm.