  Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. Perfect nonlinear monomials were
employed to construct optimal ternary cyclic codes with parameters $[3^m-1,
3^m-1-2m, 4]$ by Carlet, Ding and Yuan in 2005. In this paper, almost perfect
nonlinear monomials, and a number of other monomials over $\gf(3^m)$ are used
to construct optimal ternary cyclic codes with the same parameters. Nine open
problems on such codes are also presented.
