On an open problem about a class of optimal ternary cyclic codes
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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. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth. Let C_(1,e) be a cyclic code of length 3^m-1 over GF(3) with two nonzeros α and α^e, where α is a generator of GF(3^m)^* and e is a given integer. It is shown that C_(1,e) is optimal with parameters [3^m-1,3^m-1-2m,4] if one of the following conditions is met. 1) m≡0(mod 4), m≥ 4, and e=3^m/2+5. 2) m≡2(mod 4), m≥ 6, and e=3^m+2/2+5.
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