A non-trivial product of filtration s + 6 in the stable homotopy groups of spheres

By a method improving that of [1], the authors prove the existence of a nontrivial product of filtration, s + 6, in the stable homotopy groups of sphere, πt-6S, which is represented up to non-zero scalar by in the Adams spectral sequence, where p ≥ 7,n ≥ m +2 ≥ 5, q = 2(p - 1), 0 ≤ s < p -2, t = (s+2 + (s +2) p + pm + pn)q.The advantage of this method is to extend the range of s without much complicated argument as in [1].