A family of optimal ternary cyclic codes from the Niho-type exponent

Abstract

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. The objective of this paper is to present a family of optimal ternary cyclic codes from the Niho-type exponent. Specifically, for an odd integer m and a positive integer r with 4r≡1(modm), a family of cyclic codes C(u,v) of length 3m−1 over GF(3) with two nonzeros βu and βv is studied, where β is a generator of GF(3m)⁎, u=(3m+1)/2 and v=3r+2 is the ternary Niho-type exponent. The parameters of this family of cyclic codes are determined. It turns out that C(u,v) is optimal with respect to the Sphere Packing bound if 9∤m and otherwise almost optimal. Thanks to a recent proof of the Dobbertin–Helleseth–Kumar–Martin conjecture by Katz and Langevin, the dual of this family of cyclic codes is shown to have at most five nonzero weights.