On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on Fm3. It is easy to see that for all a∈Fm3. We completely characterize those a∈Fm3 for which , and . The simplicity of the characterization allows us to count the number of the a∈Fm3 belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.