On some conjectures about optimal ternary cyclic codes

Abstract

Cyclic codes are a subclass of linear codes and have efficient encoding and decoding algorithms over finite fields, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, by considering the solutions of certain equations over finite fields, one of the nine conjectures proposed by Ding and Helleseth about optimal cyclic codes in (IEEE Trans Inf Theory 59(9):5898–5904, 2013) is settled. In addition, we make progress toward other two conjectures.