A class of optimal ternary cyclic codes and their duals

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. Let m=2ℓ+1m=2ℓ+1 for an integer ℓ≥1ℓ≥1 and π   be a generator of GF(3m)⁎GF(3m)⁎. In this paper, a class of cyclic codes C(u,v)C(u,v) over GF(3)GF(3) with two nonzeros πuπu and πvπv is studied, where u=(3m+1)/2u=(3m+1)/2, and v=2⋅3ℓ+1v=2⋅3ℓ+1 is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over GF(3m)GF(3m), the cyclic code C(u,v)C(u,v) is shown to have minimal distance four, which is the best minimal distance for any linear code over GF(3)GF(3) with length 3m−13m−1 and dimension 3m−1−2m3m−1−2m according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied.